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+      "example": "# In Vitro Antiproliferative and Antioxidant Effects of Extracts from Rubus caesius Leaves and Their Quality Evaluation\n\nDaniel Grochowski, Roman Paduch, Adrian Wiater, Adrianna Dudek, Mabgorzata Pleszczynska, Monika Tomczykowa, Sebastian Granica, Paulina Polak, Michab Tomczyk\n\n## Abstract\n\nThe present study was performed to evaluate the effect of different extracts and subfractions from Rubus caesius leaves on two human colon cancer cell lines obtained from two stages of the disease progression lines HT29 and SW948. Tested samples inhibited the viability of cells, both HT29 and SW948 lines, in a concentration-dependent manner. The most active was the ethyl acetate fraction which, applied at the highest concentration (250 𝜇g/mL), decreased the viability of cells (HT29 and SW948) below 66%. The extracts and subfractions were also investigated for antioxidant activities on DPPH and FRAP assays. All extracts, with the exception of water extract at a dose of 250 𝜇g/mL, almost totally reduced DPPH. The highest Fe 3+ ion reduction was shown for the diethyl and ethyl acetate fractions. It was more than 6.5 times higher (at a dose 250 𝜇g/mL) as compared to the control. The LC-MS studies of the analysed preparations showed that all samples contain a wide variety of polyphenolics, among which ellagitannins turned out to be the main constituents with dominant ellagic acid, sanguiin H-6, and flavonol derivatives.\n\n## 1. Introduction\n\nDrugs of natural origin have been used throughout history to cure or prevent diseases. Modern phytotherapy is engaged in the production of remedies from materials derived from plants and their use in effective and safe therapy. Their main action could be aimed at three aspects: cytostatic activity, especially when therapy concerns tumour tissue, and antiinflammatory and antioxidative or free radical reduction actions. With all this in mind, we have tried to evaluate the cytotoxic and antioxidant activities of Rubus caesius extracts on two human colon cancer cell lines obtained from two stages of disease progression. Additionally, the full phytochemical profile of all the investigated extracts obtained from R. caesius leaves based on the HPLC-DAD-MS n method has been characterized for the first time. R caesius is a wellknown shrub (dewberry) extending from Europe to Siberia, but it can also be found in the United States. Folk medicine attributes many virtues to R. caesius. Further studies are required to confirm the pharmacological relevance of the findings, but now there are great expectations for its wide therapeutic application [1].\n\n## 2. Materials and Methods\n\n## 2.1. Plant Material and Preparation of Extracts and Their\n\nFractions. The leaves from wild species of R. caesius were collected during June-July 2012-2014 from Puszcza Knyszyńska, near Bialystok, Poland. A voucher specimen of plant RC-11027 has been deposited in the Herbarium of the Department of Pharmacognosy, Medical University of Białystok, Poland. All plant samples, extracts, and fractions were prepared according to previously described methods [2]. Yields are as follows: RC1, 83 mg; RC2, 79 mg; RC3, 101 mg; RC4, 9 mg; RC5, 28 mg; RC6, 96 mg. 3 Analysis. The HPLC-DAD-MS 3 analysis was performed using similar conditions described previously [2]. HPLC analyses of samples were carried out on a reversed-phase Kinetex XB-C18, 100 mm × 2.1 mm × 1.7 𝜇m column (PHENOMENEX, USA). Compounds were analysed in negative and positive ion modes (the MS 2 152, -162, and -176 amu). In the case of the detection of one of the neutral loss masses MS 3 fragmentation was performed. Analysis was carried out using scan from 𝑚/𝑧 70 to 2.200.\n\n## 2.2. HPLC-DAD-MS\n\n## 2.3. Cell Cultures.\n\nTwo human colon tumour cell lines were used. HT29 (ATCC5 HTB-386) and SW948 (ATCC CCL-2376) cell lines representing early and late stages of tumour development were cultured as monolayers in 25 mL culture flasks (NUNC, Rochester, USA). All cell lines were maintained in RPMI 1640 medium supplemented with 10% FBS (foetal bovine serum) (v/v) and antibiotics (100 U/mL penicillin, 100 𝜇g/mL streptomycin) (SIGMA, St. Louis, MO, USA) at 37 ∘ C in a humidified atmosphere with 5% CO 2.\n\n## 2.4. MTT Assay.\n\nThe MTT assay is based on the conversion of a yellow tetrazolium salt by viable cells to purple crystals of formazan. The reaction is catalysed by mitochondrial succinyl dehydrogenase. Cell sensitivity to R. caesius extracts was analysed in a spectrophotometric 3-(4,5-dimethylthiazol-2yl)-2,5-diphenyltetrazolium bromide (MTT) test according to Mosmann [3].\n\n## 2.5. Neutral Red (NR) Uptake Assay.\n\nThe NR cytotoxicity assay is based on the uptake and lysosomal accumulation of the supravital dye, Neutral Red. Dead or damaged cells do not take up the dye. The method was used as described earlier [4].\n\n2.6. Nitric Oxide (NO) Measurement. Nitrate, a stable end product of NO, was determined in culture supernatants by a spectrophotometric method based on the Griess reaction. The course of the procedure has been described previously [5].\n\n## 2.7. DPPH\n\n• Free Radical Scavenging Test. The free radical scavenging activity of extracts was analysed by the 1,1diphenyl-2-picrylhydrazyl (DPPH) assay. The test is based on the ability of antioxidants to reduce the stable dark violet radical DPPH • (SIGMA, USA) to the yellow diphenylpicrylhydrazine. The methodology has been described in our previous study [5].\n\n## 2.8. Ferric-Reducing Antioxidant Power (FRAP) Assay.\n\nThe FRAP method was used to determine the antioxidative capacity of the tested extracts. The procedure has been described earlier [4].\n\n## 2.9. Statistical Analysis.\n\nThe biological experiments were repeated three times. The data were analysed using one-way ANOVA followed by Dunnett's multiple comparison post hoc test. Only results with significance of 𝑝 ≤ 0.05 were considered significant.\n\n## 3. Results and Discussion\n\nMany species classified to the genus Rubus have been recognized as potential agents with significant effects on human health [6][7][8][9][10]. In the present work we selected leaves of blackberry R. caesius (dewberry) species traditionally used as a remedy to treat many diseases, among them gastrointestinal bleeding and diarrhoea [1,11]. More recently, Dudzińska and coauthors indicated that the extracts obtained from dewberry leaves demonstrate antiplatelet activities in whole blood, where neutrophils play a pivotal role in mediating their effects on platelets. Although these extracts do not hamper the neutrophil oxidative metabolism and do not influence the expression of neutrophil adhesive receptors, they demonstrate an ability to lower the reactive oxygen level produced by neutrophils [12]. According to the reviewed literature, little is known about the potential antiproliferative and antioxidant activity of dewberry's leaves which encouraged us to investigate this plant growing in Poland. In addition, there is no solid evidence describing the chemical composition of the species.\n\nFor the first time, we initiated a detailed phytochemical analysis of secondary metabolites and confirmed the presence of derivatives of quercetin and kaempferol, as well as ellagitannins [1,11]. The fingerprints of the analysed R. caesius extracts were established using the HPLC-DAD-MS 3 method. The analysis revealed the presence of thirty-five constituents (Figure 1) comprising ellagitannins and their derivatives, phenolic acids, as well as flavonoids. In the RC1 (water), RC2 (50% methanol), and RC3 (methanol) extracts ellagic acid [22] and sanguiin H-6 [23] were detected as the dominating constituents. The subfractions RC4 (diethyl ether), RC5 (ethyl acetate), and RC6 (n-butanol) contained a wide variety of phenolic acids [  1 contains detailed UV-Vis and MS data for all the detected compounds together with their preliminary or full identification. These phytoconstituents express reductive activity on free radicals and may limit the appearance of mutations or even participate in DNA repair [24]. There are a few reports concerning the antitumour activity of other Rubus leaves extracts, but no data are available supporting the extracts from dewberry leaves [15,16,22,25,26]. Previous studies on in vitro models suggest that berry from Rubus species may influence colorectal cancer cell survival in concert terms proliferation and apoptosis [17]. Komes and coworkers also revealed that infusion from R. fruticosus leaves may induce cytotoxic action against human colon cells, depending of time and concentration [26]. On the other hand, cancer development is closely associated with inflammation and mutatory microenvironments containing free radicals. In another study, a triterpenoid-rich fraction from R. coreanus has been shown to express strong antiinflammatory activity towards injured colonic tissue [27]. Therefore, we decided to aim our study at the cytotoxic and reduction activity of R. caesius leaves in human colon carcinoma cells. Studies on the biological activity of different extracts and subfractions obtained from dewberry were based on two analyses (MTT and NR assays) which were performed on two human colon tumour cell lines HT29 (Duke's A) and SW948 (Duke's C). They were selected to show the reactivity of the early stage of this tumour development. Our study revealed that the tested R. caesius extracts expressed no cytotoxic activity. Samples RC2, RC5, and RC6 in a range of concentrations up to 200 𝜇g/mL induced mitochondrial action about 20% above the HT29 cells control (Figure 2). The SW948 cell line was more sensitive to the activity of the tested samples (RC1-3, RC5) than the HT29 line. The inhibitory effect was concentration-dependent. On the other hand, samples RC4 and RC6 induced succinyl dehydrogenase activity. The tested extracts inhibited, in a concentrationdependent manner, the viability of cells of both HT29 and SW948 lines. The most active was RC5, which, applied at the highest concentration (250 𝜇g/mL), decreased the viability of cells (HT29 and SW948) below 66% (Figure 3). The butanolic fraction (RC6), which at the highest concentration did not decrease the viability of cells below 87%, was less active. Our tests revealed that R. caesius extracts possess reductive activity. All extracts, with the exception of RC1 at a dose of 250 𝜇g/mL, almost totally reduced DPPH (Figure 4). RC1 at this concentration reduced only half of the radical. Fractions RC4 and RC5 at the lowest concentrations used expressed strong antioxidative activity. IC 50 values of the tested samples' activity and their comparison to the Trolox action are presented in Table 2. Antioxidant activity of the selected extracts was also determined by the FRAP method, which is based on the analysis of the Fe 3+ ions reduction ability of the tested compounds. The highest Fe 3+ ion reduction was shown for the RC4 and RC5 extracts. It was more than 6.5 times higher (at a dose 250 𝜇g/mL) as compared to the control (Figure 5). This result is comparable to 155 𝜇g/mL of ascorbic acid reductive activity. Similarly to DPPH, the lowest reductive action was shown for the RC1 extract. At the highest concentration applied, its activity was only 2.2 times (activity corresponding to 53 𝜇g/mL of ascorbic acid) stronger than the control.\n\nWe showed that the tested extracts (RC1-RC3) and subfractions (RC4-RC6) decreased the viability of cells, acting cytotoxically on tumour cells, and simultaneously expressed strong reductive activity. Our results are in agreement with studies by Durgo et al. [25], showing that red raspberry leaf extract expresses cytotoxic and antioxidative effects in the human colon adenocarcinoma (SW480) cell line. This activity was assigned mainly to polyphenolic compounds present in the plant material. Our results also confirmed results of Dai and coauthors, who revealed that the extracts from blackberry significantly limited HT29 human colon tumour cells growth, and the effect was dependent on the concentration applied. This effect was closely connected with the high content of anthocyanins [22]. Moreover, it was shown that acetone extract of R. fairholmianus roots influenced human colon tumour cell morphology and reduced their viability via limitation of the intracellular ATP pool and changes in cells' metabolic activity. As a consequence, depleted ATP quantity decreased the tumour cell proliferation rate and stimulated their death, mainly in the apoptotic pathway [28]. Furthermore, extracts from lyophilized fruits of R. occidentalis may modulate host immune system processes by impacting on the function and viability of activated human  CD4+ and CD8+ T lymphocytes [13]. It may indirectly influence tumour cell development and further metastasis. Analyses in this immunological direction have also been expanded with the use of R. coreanus extracts loaded in gelatin nanoparticles. They were used as transport vehicles for the plant extracts and resulted in the significant enhancement of T, B, and NK cells' functionality in all areas of their immune activity [14]. Interesting results using cold water extracts of fresh fruits of R. caesius were shown by Turker et al. [29], who found a 100% antitumour efficiency of these extracts on cancer cells. In another study, similar effect was also supported by its antioxidative action with a relatively low IC 50 value of 5 𝜇g/mL [19]. Similarly, the shoot extracts of R. idaeus were found to be a source of sanguiin H-6 and ellagic acid, which exhibit antioxidative as well as cytotoxic activity [18]. Lee and coauthors additionally revealed that sanguiin H-6 induces morphological changes in tumour cells which are similar to apoptotic features. However, this compound  does not affect the cancer cell cycle. In general, the molecular pathway of sanguiin H-6 activity is mediated by MAPK p38 and BID cleavage with the participation of caspase-8 [20]. Kim and coworkers have shown that the aqueous extract of the incompletely ripened fruit of R. coreanum inhibits cell proliferation and stimulates apoptosis in HT29 cells and that this may be mediated by its ability to activate the caspase-3 pathway [30]. In other study, Bowen-Forbes and coworkers showed that fruit extracts obtained from some Rubus species also exhibited great potential to inhibit colon, breast, lung, and gastric cancer cell growth. The authors speculate that the anticancer effect may partially depend on inhibitory action on cyclooxygenase-2 (COX-2) functionality. Moreover, due to high anthocyanin content, it may also strongly influence the oxidative condition in the tumour cell microenvironment [23].\n\nBesides the general activity of Rubus extracts on tumour cells, it was shown that the range of such action is based on the horticultural parameters of the plant material. Production factors, both genetic and environmental, determine the usefulness of plants as a material for specific destiny, for example, chemoprevention. Therefore, the degree of inhibition of human colon tumour cell proliferation depends not only on general active phytoconstituents presence, but also on their specific composition which is dependent on cultivar, production site, or stage of maturity [21].\n\nGenerally, plant extracts have many biological activities directly aimed at cell morphology and proliferation or indirectly by possessing reductive feature which influence inflammatory state modulating immune system reactivity. In our study R. caesius leaf extract revealed tumour cell growth limiting activity, on both the morphology and metabolism levels. Moreover, its antioxidative activity may be connected with colon origin tumour cells growth reduction.\n\n## References\n\n1. Rejewska, Sikora, Tomczykowa et al. (2013) \"Rubus caesius\" *Pharmacognosy Communications*\n\n2. Tomczyk, Pleszczyńska, Wiater et al. (2013) \"In vitro anticariogenic effects of Drymocallis rupestris extracts and their quality evaluation by HPLC-DAD-MS 3 analysis\" *Molecules*\n\n3. Mosmann (1983) \"Rapid colorimetric assay for cellular growth and survival: application to proliferation and cytotoxicity assays\" *Journal of Immunological Methods*\n\n4. Paduch, Woźniak, Niedziela et al. (2014) \"Assessment of eyebright (Euphrasia officinalis L.) extract activity in Evidence-Based Complementary and Alternative Medicine relation to human corneal cells using in vitro tests\" *Balkan Medical Journal*\n\n5. Paduch, Woźniak (2015) \"The effect of Lamium album extract on cultivated human corneal epithelial cells (10.014 pRSV-T)\" *Journal of Ophthalmic and Vision Research*\n\n6. Patel, Rojas-Vera, Dacke (2004) \"Therapeutic constituents and actions of Rubus species\" *Current Medicinal Chemistry*\n\n7. Rocabado, Bedoya, Abad et al. (2008) \"Rubus-a review of its phytochemical and pharmacological profile\" *Natural Product Communications*\n\n8. Holst, Haavik, Nordeng (2009) \"Raspberry leaf-should it be recommended to pregnant women?\" *Complementary Therapies in Clinical Practice*\n\n9. Gouveia-Figueira, Castilho (2015) \"Phenolic screening by HPLC-DAD-ESI/MS n and antioxidant capacity of leaves, flowers and berries of Rubus grandifolius Lowe\" *Industrial Crops and Products*\n\n10. Li, Du, He (2015) \"Chemical constituents and biological activities of plants from the genus Rubus\" *Chemistry & Biodiversity*\n\n11. Gudej, Tomczyk (2004) \"Determination of flavonoids, tannins and ellagic acid in leaves from RubusL. species\" *Archives of pharmacal research*\n\n12. Dudzinska, Bednarska, Boncler et al. (2016) \"The influence of Rubus idaeus and Rubus caesius leaf extracts on platelet aggregation in whole blood. Cross-talk of platelets and neutrophils\" *Platelets*\n\n13. Mace, King, Ameen (2014) \"Bioactive compounds or metabolites from black raspberries modulate T lymphocyte proliferation, myeloid cell differentiation and Jak/STAT signaling\" *Cancer Immunology and Immunotherapy*\n\n14. Seo, Choi, Lee (2011) \"Enhanced immunomodulatory activity of gelatin-encapsulated Rubus coreanus Miquel nanoparticles\" *International Journal of Molecular Sciences*\n\n15. George, Parimelazhagan, Kumar et al. (2015) \"Antitumor and wound healing properties of Rubus ellipticus Smith\" *Journal of Acupuncture and Meridian Studies*\n\n16. Zhang, Lu, Jiang et al. (2015) \"Bioactivities and extraction optimization of crude polysaccharides from the fruits and leaves of Rubus chingii Hu\" *Carbohydrate Polymers*\n\n17. Brown, Gill, Mcdougall et al. (2012) \"Mechanisms underlying the anti-proliferative effects of berry components in in vitro models of colon cancer\" *Current Pharmaceutical Biotechnology*\n\n18. Krauze-Baranowska, Głód, Kula (2014) \"Chemical composition and biological activity of Rubus idaeus shootsa traditional herbal remedy of Eastern Europe\" *BMC Complementary and Alternative Medicine*\n\n19. Conforti, Marrelli, Carmela (2011) \"Bioactive phytonutrients (omega fatty acids, tocopherols, polyphenols), in vitro inhibition of nitric oxide production and free radical scavenging activity of non-cultivated Mediterranean vegetables\" *Food Chemistry*\n\n20. Lee, Ko, Kim (2016) \"Inhibition of A2780 human ovarian carcinoma cell proliferation by a Rubuscomponent, sanguiin H-6\" *Journal of Agricultural and Food Chemistry*\n\n21. Johnson, Bomser, Scheerens et al. (2011) \"Effect of black raspberry (Rubus occidentalis L.) extract variation conditioned by cultivar, production site, and fruit maturity stage on colon cancer cell proliferation\" *Journal of Agricultural and Food Chemistry*\n\n22. Dai, Patel, Mumper (2007) \"Characterization of blackberry extract and its antiproliferative and anti-inflammatory properties\" *Journal of Medicinal Food*\n\n23. Bowen-Forbes, Zhang, Nair (2010) \"Anthocyanin content, antioxidant, anti-inflammatory and anticancer properties of blackberry and raspberry fruits\" *Journal of Food Composition and Analysis*\n\n24. Rajendran, Ho, Williams et al. (2011) \"Dietary phytochemicals, HDAC inhibition, and DNA damage/repair defects in cancer cells\" *Clinical Epigenetics*\n\n25. Durgo, Belščak-Cvitanović, Stančić et al. (2012) \"The bioactive potential of red raspberry (Rubus idaeus L.) leaves in exhibiting cytotoxic and cytoprotective activity on human laryngeal carcinoma and colon adenocarcinoma\" *Journal of Medicinal Food*\n\n26. Komes, Belščak-Cvitanović, Ljubičić (2014) \"Formulating blackberry leaf mixtures for preparation of infusions with plant derived sources of sweeteners\" *Food Chemistry*\n\n27. Shin, Cho, Choi (2014) \"Anti-inflammatory effect of a standardized triterpenoid-rich fraction isolated from Rubus coreanus on dextran sodium sulfate-induced acute colitis in mice and LPS-induced macrophages\" *Journal of Ethnopharmacology*\n\n28. George, Tynga, Abrahamse (2015) \"In vitro antiproliferative effect of the acetone extract of Rubus fairholmianus gard. Root on human colorectal cancer cells\" *BioMed Research International*\n\n29. Turker, Yildirim, Karakas (2012) \"Antibacterial and antitumor activities of some wild fruits grown in Turkey\" *Biotechnology and Biotechnological Equipment*\n\n30. Kim, Lee, Shin et al. (2005) \"Induction of apoptosis by the aqueous extract of Rubus coreanum in HT-29 human colon cancer cells\" *Nutrition*<|endoftext|>"
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+      "example": "# Large-Scale Sequence Analysis of M Gene of Influenza A Viruses from Different Species: Mechanisms for Emergence and Spread of Amantadine Resistance ᰔ †\n\nYuki Furuse, Akira Suzuki, Hitoshi Oshitani\n\n## Abstract\n\nInfluenza A virus infects many species, and amantadine is used as an antiviral agent. Recently, a substantial increase in amantadine-resistant strains has been reported, most of which have a substitution at amino acid position 31 in the M2 gene. Understanding the mechanism responsible for the emergence and spread of antiviral resistance is important for developing a treatment protocol for seasonal influenza and for deciding on a policy for antiviral stockpiling for pandemic influenza. The present study was conducted to identify the existence of drug pressure on the emergence and spread of amantadine-resistant influenza A viruses. We analyzed data on more than 5,000 virus sequences and constructed a phylogenetic tree to calculate selective pressures on sites in the M2 gene associated with amantadine resistance (positions 26, 27, 30, and 31) among different hosts. The phylogenetic tree revealed that the emergence and spread of the drug-resistant M gene in different hosts and subtypes were independent and not through reassortment. For human influenza virus, positive selection was detected only at position 27. Selective pressures on the sites were not always higher for human influenza virus than for viruses of other hosts. Additionally, selective pressure on position 31 did not increase after the introduction of amantadine. Although there is a possibility of drug pressure on human influenza virus, we could not find positive pressure on position 31. Because the recent rapid increase in drug-resistant virus is associated with the substitution at position 31, the resistance may not be related to drug use.\n\nInfluenza virus, a common cause of respiratory infections worldwide, infects humans and avian, swine, and equine species. The virus has a negative-sense, single-stranded RNA genome, which is comprised of eight segments that comprise 12 genes (42). Influenza A viruses cause epidemics and pandemics by antigenic drift and antigenic shift, respectively (42). Antigenic drift is due to an accumulation of point mutations leading to minor and gradual antigenic changes. Antigenic shift involves major antigenic changes by introduction of new hemagglutinin (HA) and/or neuraminidase subtypes into the human population. Since the majority of humans do not have immunity to such novel subtypes, the morbidity and mortality impacts of pandemic influenza can be much higher than those of seasonal influenza.\n\nAmantadine and rimantadine are antiviral agents used for influenza A infection. Both inhibit virus replication by blocking the acid-activated ion channel formed by the virion-associated M2 protein encoded by the M gene (41). The M gene (1,027 bp) encodes two proteins, M1 (at nucleotide positions 26 to 784) and M2 (at positions 26 to 51 and 740 to 1007) (23). The M2 protein comprises 97 amino acids and has ion channel activity (27). Mutations of the M2 gene associated with amantadine (and rimantadine) resistance include mutations at amino acid positions 26, 27, 30, 31, and 34 (1,12). Amantadineresistant strains of influenza A virus are commonly isolated from clinical samples (14,34), and they can be generated easily in vitro by culturing the viruses in the presence of amantadine (3,12). Resistant strains can replicate as efficiently as sensitive ones, and they can also transmit efficiently (1,13). Recently, a significant worldwide increase in resistant strains has been reported, not only among seasonal influenza viruses in humans (H1N1 and H3N2) (5,31), but also in H5N1 avian influenza viruses (7,15). Most of the resistant strains have a serine-toasparagine substitution at amino acid position 31 (S31N) in the M2 gene. However, controversies exist regarding the implication of drug pressure (i.e., increasing use of the drug) in increasing resistance (5,10,17,30,35). It is believed that excess use of amantadine leads to an increase in amantadine-resistant viruses (5,17,30), but the drug pressure alone may not be able to explain the recent rapid and significant increase in amantadine resistance (10,35), and such resistance may be totally unrelated to increasing drug use. For example, oseltamivirresistant influenza A virus (H1N1) is increasing worldwide (22), yet it is unclear if the increasing oseltamivir resistance is associated with use of the drug, since resistance emerged in northern Europe, where oseltamivir is not widely used (11,20). Understanding the mechanism responsible for the emergence and spread of antiviral resistance is important for developing a treatment protocol for seasonal and pandemic influenza.\n\nWe conducted the present study to clarify whether drug pressure affects the evolution of the M gene by analyzing large numbers of sequences of influenza A viruses from different hosts. The main purpose of the study was to understand the emergence and spread of amantadine resistance among different hosts.\n\n## MATERIALS AND METHODS\n\nSequence data. All data were obtained from the influenza virus sequence database (Influenza Virus Resource [http://www.ncbi.nlm.nih.gov/genomes/FLU /FLU.html], accessed on 21 July 2008 [4]). All sequencing data for the strains with a full-length M gene and any subtypes of influenza A virus from different host species, including avian, canine, equine, human, and swine viruses, were included. Sequences derived from laboratory strains and different sequences from the same strains, verified by the strain name, were excluded. A total of 5,489 sequences were obtained (the accession numbers are listed in the supplemental material). The sequences containing ambiguous nucleotides, minor insertions, minor deletions (data for full-length coding regions were used), or premature termination codons were excluded. As a result, 5,060 sequences were used for analysis. The sequencing data were obtained together with information about the host, subtype, isolation year, and isolation location. The numbers of sequences of viruses in each host are given in Table 1. A multiple-sequence alignment of the nucleotide sequences, which did not contain any gaps, was constructed using ClustalW. Among all 5,060 sequences, the number of strains with the amantadine resistance mutation was determined.\n\nPhylogenetic-tree analysis. A phylogenetic tree was inferred by RAxML (37) with all 5,060 sequences. The data used were the sequences for the coding region only, i.e., at nucleotide positions 26 to 1007. The basic sequential algorithm of RAxML is outlined elsewhere (8). RAxML is one of the fastest and most accurate sequential phylogeny programs (38). In this method, a rapid bootstrap search was combined with a rapid maximum-likelihood search of the original alignment. The tree was constructed using the Web server RAxML BlackBox (http://phylobench.vital-it.ch/raxml-bb/) (37). The M genes with amantadine resistance mutations were colored by FigTree (version 1.1.2).\n\nData sets for each influenza virus host. Data sets for each host (avian, canine/ equine, human, and swine) were constructed. Only sequences from the hostspecific lineage in the phylogenetic tree were used. For example, the data set for the human influenza viruses consists of sequences of human influenza A viruses that are in the human lineage in the phylogenetic tree (Fig. 1a). H5N1 influenza viruses that infect humans were excluded from the analyses because humans were accidental hosts infected with viruses in an avian lineage. These accidental infections should not reflect host-specific evolution. Also, sequences with identical nucleotides in the same data set were removed because the data set should not include identical sequences to analyze selective pressure. The profile of sequences we analyzed is in Table S1 in the supplemental material.\n\nThe number of base substitutions per site from averaging over all sequence pairs was calculated to define the diversity of sequences in a data set (see Tables S1 andS2 in the supplemental material) using the maximum composite likelihood method in MEGA (ver. 4) (21).\n\nEvaluation of pressure. Selective pressures were calculated for each data set for each influenza virus host. Phylogenetic trees for each data set were constructed with the maximum-likelihood method implemented in PhyML-aLRT (2) using the General Time Reversible model (four rate categories, with all parameters estimated from the data).\n\nSelective pressure among host populations was calculated using the trees. Selective pressure was analyzed with HyPhy (29). All analyses in HyPhy were conducted after identifying the best-fit model out of every possible time-reversible model (e.g., F81 and HKY85) by Akaike's information criterion (24,33).\n\nRelative rates of nonsynonymous (dN) and synonymous (dS) substitutions were calculated. Positive selection sites in human influenza virus were detected by two methods, single-likelihood ancestor counting (SLAC) and fixed-effects likelihood (FEL). The relative rates of nonsynonymous and synonymous substitutions were compared. Sites where dN/dS was Ͼ1 and where dN/dS was Ͻ1 were inferred as positively and negatively selected, respectively. The details of the two methods are described elsewhere (6,19,33). Briefly, in the SLAC method, the nucleotide and codon model parameter estimates are used to reconstruct the ancestral codon sequences at internal nodes of the tree. The single most likely ancestral sequences are then fixed as known variables and applied to infer the expected number of nonsynonymous or synonymous substitutions that have occurred along each branch for each codon position. SLAC is a substantially modified and improved derivative of the Suzuki-Gojobori method (40). The FEL method is based on maximum-likelihood estimates. The FEL method estimates the ratio of nonsynonymous to synonymous substitutions on a site-by-site basis for the entire tree (eFEL) or only the interior branch (iFEL). iFEL is essentially the same as eFEL, except that selection is tested only along internal branches of the phylogeny (28). Separate analyses were conducted by testing hypotheses for the entire tree, the internal branch, and the terminal branch: the SLAC (for the entire tree [eSLAC], internal branches [iSLAC], and terminal branches [tSLAC]) and FEL (for the entire tree [eFEL] and internal branches [iFEL]) methods. Pond et al. (28) revealed that many recent nonsynonymous substitutions, i.e., those in the terminal branches of the tree, were not represented on internal branches. At codons where internal substitutions are seen, the strength of selection along terminal branches is high.\n\nComparison of pressures. The differential of evolutionary pressures was analyzed by HyPhy. HyPhy tests whether the dN/dS ratios at a given site differ between two data sets along the entire tree (eFEL) or only with interior sequences (iFEL). The details are described elsewhere (28,33). The differential between hosts was tested. In addition, the human data set that was constructed as described above was divided by time (before 1965 and after 1966, and before 1999 and after 2000) to create new datasets (see Table S2 in the supplemental material). The differentials between them were also tested.\n\n## RESULTS\n\nFrequency of drug resistance. The numbers of strains with amantadine resistance mutation(s) among different hosts are shown in Table 1. There were no amantadine resistance mutations at position 34 of M2 in any of the strains except laboratory strains. We therefore conducted the analyses focusing on positions 26, 27, 30, and 31 as sites for amino acid substitutions associated with naturally occurring amantadine resistance. Canine/equine influenza virus had no amantadine-resistant strains. Amantadine-resistant mutations were detected at all four sites (positions 26, 27, 30, and 31) in human influenza virus, as well as avian and swine influenza viruses. We also found strains with double resistance mutations (positions 27 and 31) in these hosts. Amantadine resistance mutations were detected most frequently at position 31, followed by position 27, in all hosts except canine/equine. We found more amantadine-resistant strains in human influenza virus after 2000 (Table 1) because of an increase in the resistance mutation at position 31 in both H3N2 and H1N1. Since 2000, amantadineresistant strains of avian influenza virus have also been found more frequently, and many of these (170 out of 227) were H5N1 viruses. The most common resistance mutation was S31N in human, avian, and swine influenza viruses.\n\nPhylogenetic trees. Phylogenetic trees for all M gene sequence data are shown in Fig. 1a. The features of the tree were described in detail under subtypes, hosts, and temporal and geographical distribution in our previous study (9). The analysis revealed seven host-specific lineages: (i) a human influenza virus lineage that consisted of H1N1 between 1918 and 1954, H2N2 between 1957 and 1967, and H3N2 after 1968; (ii) a human influenza virus lineage that comprised H1N1 after 1977; (iii) an avian lineage that included viruses mainly from Asia, but also from other regions; (iv) an avian lineage that included viruses mostly from North America; (v) a swine lineage that was between the human and avian lineages and mainly included viruses from North America; (vi) a swine lineage that diverged from an avian lineage and consisted of swine viruses after 1980, mainly from Europe; and (vii) a canine/equine lineage that diverged from an avian-lineage root.\n\nStrains with mutations associated with amantadine resistance were identified in all lineages except the canine/equine lineage (Fig. 1b). Amantadine resistance mutations appeared across different subtypes, different hosts, and different geographic regions.\n\nThe viruses with mutations at positions 26, 27, and 30 were found sporadically, but those viruses did not become dominant strains and disappeared. There were eight major clusters of resistant strains (indicated in Fig. 1b). All of these major resistant clusters had the S31N mutation. Such a cluster was found in the 1930s (human H1N1 1933 in Fig. 1b). They were not laboratory strains (e.g., A/Melbourne/35, CY009325, and A/Alaska/1935, CY019956). Most major resistant clusters were found after the late 1990s.\n\nSite-by-site pressures for human influenza virus. We analyzed selective pressures on human influenza virus. The data on the M2 sites associated with amantadine resistance are shown in Table 2. \"dN/dS\" indicates the ratio of nonsynonymous and synonymous substitutions at each codon. When the pressure on a codon is significantly larger than 1, the site is regarded as under significant positive selection. When the pressure on a codon is significantly smaller than 1, the site is regarded as under significant negative selection (33,40).\n\nThe selective pressure on the entire sequence of M2 was 0.45. Position 26 had a dN/dS ratio greater than 1 (1.35), but it was not statistically significant (P Ͼ 0.05 for all tests). Position 27 had a much larger dN/dS ratio (4.42) and was under significant positive selection based on tSLAC (P ϭ 0.039). Position 30 had a dN/dS ratio smaller than 1, though significant negative selection was not found in any tests. Position 31 showed results similar to those of position 30.\n\nDifferences in selective pressures between hosts. We analyzed the differences in selective pressures on sites associated with amantadine resistance between hosts. The site-specific selective pressures (dN/dS) for each host are shown in Fig. 2a. The dN/dS ratio of avian influenza virus was calculated, excluding sequences of chicken viruses (discussed below).\n\nWe found higher selective pressures for human viruses than for viruses of the other hosts at positions 26 and 31 (Fig. 2a). Significantly higher selective pressures for human viruses were found only when compared to avian viruses at position 26 and to canine/equine viruses at position 27. The significant differences were observed only by a test of the entire tree (eFEL) (Fig. 2a). iFEL, which is a test for internal branches, did not detect significant differences (see Table S3 in the supplemental material). We could not find any significant differences between human and swine viruses. Furthermore, human influenza virus was under lower selective pressure at positions 27 and 30 than the avian virus, though the difference was not significant.\n\nChange in selective pressure with time. We divided the human data set by the year of isolation: before 1965 versus after 1966 and before 1999 versus after 2000. In 1966, amantadine was approved as a drug for influenza virus infection in the United States (25), and in 2000, the escalating trend of circulating amantadine-resistant viruses in humans began (5). We found that the entire selective pressure for the M gene (both M1 and M2) became smaller with time (data not shown).\n\nThe dN/dS ratio for positions 26, 27, and 30 associated with amantadine resistance has increased since 1966. The selective pressures were also higher after 2000 than before 1999 (Fig. 2b), although there were no significant differences. In contrast, the dN/dS ratio for position 31 became smaller rather than larger in both analyses (before 1965 versus after 1966 and before 1999 versus after 2000) (Fig. 2b).\n\n## DISCUSSION\n\nAmantadine-resistant strains were found in avian and swine, as well as human, influenza viruses. The recent spread of amantadine-resistant viruses is caused by the emergence of FIG. 2. Differentials of selective pressures on sites associated with amantadine resistance. (a) Differentials of selective pressures on sites associated with amantadine resistance between hosts. Selective pressures for human viruses are higher than for the viruses of other hosts at positions 26 and 31. Significantly higher selective pressures for human viruses were found only when compared to avian viruses at position 26 and only when compared to canine/equine viruses at position 27. The significant differences were observed only by eFEL. No significant differences were found by iFEL. The dN/dS ratios at positions 26 and 30 for canine/equine influenza virus could not be calculated because both the denominator and the numerator were zero. The dN/dS ratios at positions 27 and 31 for canine/equine influenza viruses are zero, as only the numerators were zero. *, significant differences were found by eFEL (P Ͻ 0.05). (b) Differentials of selective pressures on sites associated with amantadine resistance in human influenza virus by time. The dN/dS ratios for positions 26, 27, and 30 have become larger since 1966 (the introduction of amantadine) and 2000 (the beginning of the recent surge of amantadine-resistant strains), though there are no significant differences. In contrast, the dN/dS ratio for position 31 became smaller rather than larger. \"inf\" means infinity, as the denominator was zero. The dN/dS ratios at position 26 before 1965, at position 30 before 1965, and at position 30 before 1999 were zero, as only the numerators were zero. No significant differences were found by either eFEL or iFEL. viruses with the S31N mutation in the M2 gene. The phylogenetic tree suggests that viruses with an amantadine resistance mutation(s) occurred independently by point mutation in the M gene in each lineage (Fig. 1). The emergence of the S31N mutation was not caused by acquiring the M gene from other hosts/subtypes by reassortment.\n\nThe tree shows that all major resistant clusters have the S31N mutation. Only M genes with the S31N mutation were maintained and could become dominant strains, indicating that strains with this mutation could efficiently transmit it to the next generation. It must be noted that strains with S31N appeared and were maintained in the human population in the 1930s, which was before amantadine was discovered and used (Fig. 1b). Furthermore, it is intriguing that clusters of M genes with mutations at position 31 have emerged separately since the late 1990s in different hosts and different subtypes (Fig. 1b). The influenza viruses of various hosts acquired the mutation at position 31 independently and almost simultaneously. Schmidtke et al. reported the emergence of amantadine-resistant strains of swine influenza virus in the 1980s and suggested that this might have been caused by a reassortment event and that further reassortment between these swine and human influenza viruses could cause an increase in the amantadineresistant M gene in human influenza virus (32). Our results are contrary to this suggestion. Although amantadine resistance did increase in human and avian viruses after 2000 (Table 1), these resistant viruses did not acquire the M gene from swine viruses (Fig. 1b).\n\nThe next question was why amantadine resistance has increased so rapidly. One possible explanation is that there was drug pressure on the influenza viruses of various hosts that led to an amino acid change at position 31. Otherwise, S31N could be just a genetic variant in a diverse gene pool. It should also be noted that amantadine has been less frequently used since 2000 because neuraminidase inhibitors (oseltamivir and zanamivir) were licensed and became the most commonly prescribed drugs for influenza A and B virus infections in developed countries. However, there is a possibility of localized amantadine use in some countries even after 2000 (5,35). We conducted further analysis to determine whether the drug pressure had any effect on the recent emergence and spread of viruses with the S31N mutation.\n\nAnalysis for selective pressure on human influenza virus indicates that the mutations at sites associated with amantadine resistance are not generally driven by external pressure affecting the entire tree, including drug pressure. If anything, the pressure affects only terminal branches without affecting internal branches at position 27, because only tSLAC, which is a test only for the terminal branches, found significant positive selection at position 27. iSLAC and iFEL, which are tests for internal branches, did not find significant positive selection. Although Suzuki showed that there was no positive pressure on the sites in human influenza virus (39), we found positive selection at position 27. This must be because we applied various methods of calculation and analyzed a data set that was 10 times larger than the data Suzuki used.\n\nAlthough positive selection was not found in three of the four sites linked to amantadine resistance, we could not reject drug pressure on evolution of the M gene. In fact, the selective pressures on the sites, except position 30, were higher than the selective pressure for the entire M2 gene. Particularly at position 31, other substitutions apart from S31N must be under strong negative pressure, since we found only serine and asparagine at the site. The negative pressure might conceal drug pressure. Therefore, we compared selective pressures by hosts and time.\n\nAmantadine is known to have been used to treat human disease and possibly in poultry, such as chickens (36). Even though veterinary use is possible, drug pressure on nonhuman influenza A viruses (except possibly in chickens) will not be stronger than on human viruses. If the drug pressure is exerted on M gene evolution, it must be stronger in the human population than in other hosts, even if the pressure is not significant positive selection. For the analysis, we removed the chicken virus data from the avian data set because chicken viruses may have been under selective pressure due to amantadine use. Hill et al. found positive selection at positions 27 and 31 in H5N1 avian influenza virus (16).\n\nWe found higher selective pressure for human viruses than for those of other hosts at positions 26 and 31, while human influenza virus was under lower selective pressure at positions 27 and 30 than avian virus (Fig. 2a). We could not find any significant differences between human and swine viruses, which are unlikely to be under drug pressure. In addition, significant differences in some combinations were observed only by testing the entire tree (eFEL) and not by using iFEL, which is a test for internal branches. That is, drug pressure, if any, was not strong enough to affect the interior branches. The results could not support the hypothesis that human influenza virus is under substantially higher drug pressure than viruses of other hosts.\n\nIn case of drug pressure on sites in M2, the significance of selective pressure (dN/dS) could become larger after the introduction of amantadine and/or the beginning of excess use of amantadine. It is said that the recent rapid increase of amantadine-resistant strains might be caused by excessive use of amantadine in Asian and adjacent countries, since amantadine is available as an over-the-counter formulation in those countries (5,35).\n\nWe found that recent drug pressure might be stronger than before, although there were no significant differences (Fig. 2b). These results suggest that amantadine may be exerting pressure on human influenza viruses. However, selective pressure on position 31 did not increase even after the introduction of amantadine or the surge in drug-resistant viruses. Although selective pressure on position 31 has decreased, strains with the S31N mutation have increased (Table 1), suggesting that most of the resistant strains originated from a single or a few viruses in each lineage with the S31N mutation. This hypothesis is supported by the phylogenetic tree constructed in the present study (Fig. 1b). Most of the recent resistant strains have been derived from a single or a few strains and formed small clusters.\n\nWe used a mathematical-biological approach to determine if there was any selective pressure on amino acid positions associated with amantadine resistance. We could not find significant evidence for drug pressure on position 31 in human influenza virus. Shiraishi et al. showed that S31N, and also various mutants with amantadine-resistant mutations in M2, were detected in patients under treatment with amantadine (34). The reason why only S31N, which has weak selective pressure, has spread so rapidly remains unclear.\n\nIt is possible that the S31N mutation has occurred naturally to some extent, irrespective of the use of amantadine. We showed that M genes with S31N appeared and were maintained in the 1930s before the development of amantadine. Other strains with the S31N mutation in the M2 gene also appeared sporadically in the period from the 1940s to the 1990s (Table 1). M genes with S31N might have increased by genetic drift, as in Kimura's neutral theory of molecular evolution (18,26). In this theory, mutations that are not under selective pressure and are not advantageous or disadvantageous can predominate in a population by chance. Even if conversion from amantadine sensitivity to resistance caused by the S31N mutation occurs less commonly than mutations at other sites, such as position 27, the virus may be easily maintained once it occurs. Simonsen et al. proposed that a combination of S31N in the M2 gene and some specific amino acid substitutions in the HA genes was advantageous to the virus (35). However, our previous study revealed that strains with S31N without the same substitutions in the HA gene became prevalent after further reassortment (10). It is not known whether S31N alone or S31N together with other amino acid substitutions can change the growth characteristics of the virus.\n\nIn the present study, we showed that the recent rapid increase in drug-resistant virus was associated with a substitution at position 31, but the resistance may not be related to drug pressure. Further in silico, in vitro, and in vivo studies are needed to elucidate the mechanism responsible for the recent emergence of resistant strains with the S31N mutation.\n\n## References\n\n1. Abed, Goyette, Boivin (2005) \"Generation and characterization of recombinant influenza A (H1N1) viruses harboring amantadine resistance mutations\" *Antimicrob. Agents Chemother*\n\n2. Anisimova, Gascuel (2006) \"Approximate likelihood-ratio test for branches: a fast, accurate, and powerful alternative\" *Syst. Biol*\n\n3. Appleyard (1977) \"Amantadine-resistance as a genetic marker for influenza viruses\" *J. Gen. Virol*\n\n4. Bao, Bolotov, Dernovoy et al. (2008) \"The influenza virus resource at the National Center for Biotechnology Information\" *J. Virol*\n\n5. Bright, Medina, Xu et al. (2005) \"Incidence of adamantane resistance among influenza A (H3N2) viruses isolated worldwide from 1994 to 2005: a cause for concern\" *Lancet*\n\n6. Campo, Dimitrova, Mitchell et al. (2008) \"Coordinated evolution of the hepatitis C virus\" *Proc. Natl. Acad. Sci. USA*\n\n7. Cheung, Rayner, Smith et al. (2006) \"Distribution of amantadine-resistant H5N1 avian influenza variants in Asia\" *J. Infect. Dis*\n\n8. Felsenstein (1981) \"Evolutionary trees from DNA sequences: a maximum likelihood approach\" *J. Mol. Evol*\n\n9. Furuse, Suzuki, Kamigaki et al. (2009) \"Evolution of the M gene of the influenza A virus in different host species: large-scale sequence analysis\" *Virol. J*\n\n10. Furuse, Suzuki, Kamigaki et al. (2008) \"Reversion of Influenza A (H3N2) from amantadine resistant to amantadine sensitive by further reassortment in Japan during the 2006-to-2007 influenza season\" *J. Clin. Microbiol*\n\n11. Hauge, Dudman, Borgen et al. (2009) \"Oseltamivir-resistant influenza viruses A (H1N1), Norway, 2007-08\" *Emerg. Infect. Dis*\n\n12. Hay, Wolstenholme, Skehel et al. (1985) \"The molecular basis of the specific anti-influenza action of amantadine\" *EMBO J*\n\n13. Hayden, Hay (1992) \"Emergence and transmission of influenza A viruses resistant to amantadine and rimantadine\" *Curr.t Top. Microbiolo. Immunol*\n\n14. Hayden, Sperber, Belshe et al. (1991) \"Recovery of drug-resistant influenza A virus during therapeutic use of rimantadine\" *Antimicrob. Agents Chemother*\n\n15. He, Qiao, Dong et al. (2008) \"Amantadine resistance among H5N1 avian influenza viruses isolated in Northern China\" *Antivir. Res*\n\n16. Hill, Guralnick, Wilson et al. (2008) \"Evolution of drug resistance in multiple distinct lineages of H5N1 avian influenza\" *Infect. Genet. Evol*\n\n17. Ilyushina, Govorkova, Webster (2005) \"Detection of amantadine-resistant variants among avian influenza viruses isolated in North America and Asia\" *Virology*\n\n18. Kimura (1968) \"Evolutionary rate at the molecular level\" *Nature*\n\n19. Pond, Frost (2005) \"Not so different after all: a comparison of methods for detecting amino acid sites under selection\" *Mol. Biol. Evol*\n\n20. Kramarz, Monnet, Nicoll et al. (2009) \"Use of oseltamivir in 12 European countries between 2002 and 2007-lack of association with the appearance of oseltamivir-resistant influenza A(H1N1) viruses\" *Euro Surveill*\n\n21. Kumar, Nei, Dudley et al. (2008) \"MEGA: a biologistcentric software for evolutionary analysis of DNA and protein sequences\" *Brief.n Bioinform*\n\n22. Lackenby, Thompson, Democratis (2008) \"The potential impact of neuraminidase inhibitor resistant influenza\" *Curr. Opin. Infect. Dis*\n\n23. Lamb, Lai, Choppin (1981) \"Sequences of mRNAs derived from genome RNA segment 7 of influenza virus: colinear and interrupted mRNAs code for overlapping proteins\" *Proc. Natl. Acad. Sci. USA*\n\n24. Lanave, Preparata, Saccone et al. (1984) \"A new method for calculating evolutionary substitution rates\" *J. Mol. Evol*\n\n25. Maugh (1976) \"Amantadine: an alternative for prevention of influenza\" *Science*\n\n26. Ohta, Gillespie (1996) \"Development of neutral and nearly neutral theories\" *Theor. Popul. Biol*\n\n27. Pinto, Holsinger, Lamb (1992) \"Influenza virus M2 protein has ion channel activity\" *Cell*\n\n28. Pond, Frost, Grossman et al. (2006) \"Adaptation to different human populations by HIV-1 revealed by codon-based analyses\" *PLoS Comput. Biol*\n\n29. Pond, Frost, Muse (2005) \"HyPhy: hypothesis testing using phylogenies\"\n\n30. Regoes, Bonhoeffer (2006) \"Emergence of drug-resistant influenza virus: population dynamical considerations\" *Science*\n\n31. Saito, Suzuki, Li et al. (2008) \"Increased incidence of adamantane-resistant influenza A(H1N1) and A(H3N2) viruses during the 2006-2007 influenza season in Japan\" *J. Infect. Dis*\n\n32. Schmidtke, Zell, Bauer et al. (1981) \"Amantadine resistance among porcine H1N1, H1N2, and H3N2 influenza A viruses isolated in Germany between\" *Intervirology*\n\n33. Sergei, Pond, Frost (2007) \"Estimating selection pressures on alignments of coding sequences analyses using HyPhy\"\n\n34. Shiraishi, Mitamura, Sakai-Tagawa et al. (2003) \"High frequency of resistant viruses harboring different mutations in amantadine-treated children with influenza\" *J. Infect. Dis*\n\n35. Simonsen, Viboud, Grenfell et al. (2007) \"The genesis 4462 FURUSE ET AL. ANTIMICROB. AGENTS CHEMOTHER. and spread of reassortment human influenza A/H3N2 viruses conferring adamantane resistance\"\n\n36. Sipress (2005) \"Bird flu drug rendered useless\"\n\n37. Stamatakis, Hoover, Rougemont (2008) \"A rapid bootstrap algorithm for the RAxML Web servers\" *Syst. Biol*\n\n38. Stamatakis, Ludwig, Meier (2008) \"Computing large phylogenies with statistical methods: problems and solutions\"\n\n39. Suzuki (2006) \"Natural selection on the influenza virus genome\" *Mol. Biol. Evol*\n\n40. Suzuki, Gojobori (1999) \"A method for detecting positive selection at single amino acid sites\" *Mol. Biol. Evol*\n\n41. Wang, Takeuchi, Pinto et al. (1993) \"Ion channel activity of influenza A virus M2 protein: characterization of the amantadine block\" *J. Virol*\n\n42. Webster, Bean, Gorman et al. (1992) \"Evolution and ecology of influenza A viruses\" *Microbiol. Rev*\n\n43. (2009) \"LARGE-SCALE SEQUENCE ANALYSIS OF AMANTADINE RESISTANCE 4463\"<|endoftext|>"
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+      "example": "# A Survey on QoE-oriented Wireless Resources Scheduling\n\nIvo Sousa, Maria Paula Queluz, António Rodrigues\n\n## Abstract\n\nFuture wireless systems are expected to provide a wide range of services to more and more users. Advanced scheduling strategies thus arise not only to perform efficient radio resource management, but also to provide fairness among the users. On the other hand, the users' perceived quality, i.e., Quality of Experience (QoE), is becoming one of the main drivers within the schedulers design. In this context, this paper starts by providing a comprehension of what is QoE and an overview of the evolution of wireless scheduling techniques. Afterwards, a survey on the most recent QoE-based scheduling strategies for wireless systems is presented, highlighting the application/service of the different approaches reported in the literature, as well as the parameters that were taken into account for QoE optimization. Therefore, this paper aims at helping readers interested in learning the basic concepts of QoE-oriented wireless resources scheduling, as well as getting in touch with its current research frontier.\n\n## 1. Introduction\n\nWireless resources scheduling comprises the allocation of physical radio resources among users and the determination of the users' serving order (also known as prioritization). The goal is to fulfill some service requirements such as fairness (including avoiding greedy users, where one user consumes all or almost all system resources) or congestion, along with other constrains like delay or packet loss rate.\n\nCompared to wired networks, wireless channels have time-varying behaviors, hence more complex scheduling schemes are required for the latter. However, since the scheduling process allows to save resources, wireless schedulers play a crucial role in efficient management of scarce radio resources.\n\nIn order to improve the service level, wireless systems have adopted in the past years schedulers that provide Quality of Service (QoS), i.e., the network's capability to guarantee a certain level of performance to a data flow. Since QoS is usually evaluated in terms of delay, packet loss rate, jitter or throughput, QoS can be regarded as a service quality characterization that is network-centric.\n\nDespite the popularity of QoS-oriented schedulers design, the end-users -humans -have the decisive judgment about the received service quality. In literature, some pioneering authors (Khan et al., 2007;Saul, 2008;Saul and Auer, 2009;Thakolsri et al., 2009) showed that the application of a subjective-based approach may lead to significant improvements on user perceived quality, i.e., Quality of Experience (QoE), compared to network-centric approaches, such as maximization of the system throughput (i.e., the sum of the data rates that are delivered to all terminals). Hence, a shift from QoS-to QoE-oriented mechanisms design has been observed in recent years.\n\nQoE is a concept that tries to cover everything that a user experiences when dealing with multimedia services and systems (Brunnström et al., 2013); it takes into account not only the usability of a multimedia service or system, but also the information content. Consequently, QoE can be regarded as a user-centric characterization of the service quality.\n\nAs the number of dimensions involved in the users' subjective evaluation is immense, QoE-based techniques are becoming progressively more complex and sophisticated than the previous QoS-oriented algorithms. Schedulers that make use of QoE features consequently try to directly reflect the subjective experiences of the users, resulting in their resource allocation and prioritization techniques to be more efficient in terms of satisfying the users than the schedulers that adopt conventional metrics. This efficiency can be achieved by avoiding wasting resources in situations where there is a small or even no impact on the user experience. Therefore, QoE-oriented wireless resources schedulers aim to fulfill the mobile system users expectations: watch/listen what I want, anywhere, anytime.\n\nConsidering the existing literature, it was recognized a lack of a proper comprehensive guide regarding wireless schedulers design that take QoE into account (cf. Table 1): some works survey QoE models and assessment methods for a variety of services, but they do not consider QoE pro-\n\n## Focus References Remarks\n\nQoE estimation for different types of services Chikkerur et al. (2011); Lin and Kuo (2011); Chen et al. (2015c); Juluri et al. (2016); Tsolkas et al. (2017) These works address QoE assessment models for several applications (e.g., video streaming, conversational voice, web browsing, file download), but disregard QoE provisioning methodologies.\n\nQoE challenges with respect to mobile networks Baraković and Skorin-Kapov (2013); Siris et al. (2014); Liotou et al. (2015); Zhang et al. (2018) QoE monitoring and optimization issues are considered, although without fully addressing the scheduling of wireless resources. visioning algorithms; other works focus on mobile networks and provide insights on QoE management issues, yet they do not perform an in-depth study of wireless resources schedulers; finally, some surveys addressed precisely the scheduling of wireless resources, but no QoE-aware procedures were reviewed. Accordingly, this survey paper aims at filling this gap by giving an extensive overview of the key facets of QoE-oriented wireless resources scheduling. Its main contributions are:\n\n• A taxonomy and classification of approaches for QoEoriented resource scheduling in wireless networks;\n\n• A survey of existing work, including the classification according to the aforementioned taxonomy;\n\n• A brief discussion of each approach, giving the readers an idea about which parts of existing literature might be of interest to their requirements.\n\nTherefore, this survey serves as a reference for those who want to implement QoE-aware wireless resource schedulers and also aims to be a valuable contribution for those who want to perform research within this topic. This paper is organized as follows (cf. Fig. 1). The first two sections that follow this introduction provide a contextualization regarding what is QoE and how traditional schedulers work: in Section 2, the factors that influence the QoE in multimedia services over communication systems are presented, along with some QoE estimation methods; Section 3 illustrates some scheduling algorithms, ranging from the simplest ones to QoS-aware approaches, followed by the introduction of QoS-QoE mapping strategies and utility-based optimization. Section 4 provides the main contribution of this survey, namely the presentation of recent research directions regarding QoE-oriented wireless resources scheduling -state-of-the-art QoE-aware scheduling methods are discussed and classified based on the adjustments required, at the end-user devices, in order to implement the different scheduling strategies on wireless systems. Some of the important open challenges and future research opportunities are discussed in Section 5. Finally, Section 6 concludes the paper.\n\n## 2. Understanding QoE\n\nThe Qualinet white paper on definitions of QoE states that \"QoE is the degree of delight or annoyance of the user of an application or service. It results from the fulfillment of his or her expectations with respect to the utility and/or enjoyment of the application or service in the light of the user's personality and current state\" (Brunnström et al., 2013).\n\nAs far as communication systems are concerned, QoE can be affected by factors such as multimedia content, application, service, end-user device, network and context of use. For instance, Sumby and Pollack (1954) showed that if supplementary visual observation of the speaker's facial and lip movements are also utilized besides the oral speech, higher levels of noise interference can be tolerated by humans than if no visual factors were taken into ac- count. Hence, QoE assessment operations are performed in a broader domain when compared to QoS measurements -cf. Fig. 2.\n\nThe following subsections provide some details about the factors that influence the user experience, as well as some QoE estimation methods regarding multimedia services over communication systems.\n\n## 2.1. Factors Influencing QoE\n\nAccording to the Qualinet white paper (Brunnström et al., 2013), an influence factor is defined as \"any characteristic of a user, system, service, application, or context whose actual state or setting may have influence on the QoE for the user\". Factors influencing QoE may be categorized as human, system, and context factors.\n\n## 2.1.1. Human factors\n\nThe characteristics of the users such as gender, age, and visual and auditory acuity are examples of human physical factors that may impact the users' perceived quality (Laghari and Connelly, 2012). On the other hand, more variant factors such as motivation, attention level, or users' mood, i.e., emotional factors, also play an important role when addressing the QoE influence factors (Wechsung et al., 2011). Moreover, even educational background, occupation, and nationality will affect the QoE (Zhu et al., 2015b). In short, human factors that influence the perceived quality are complex and strongly interrelated, and their assessment should also take into consideration the time-dynamic perception of a service (i.e., the memory effect), where previous experiences also have influence on the current QoE (Hoßfeld et al., 2011).\n\n## 2.1.2. System factors\n\nThe technology employed for multimedia content transmission may introduce distortions or impairments in the content, which may affect the users' QoE. First, the original data need to be compressed, so that the multimedia content can be transmitted through a capacity-limited network. This encoding process, which incorporates many technical decisions such as the chosen bitrate (constant or variable), video frame rate or spatial resolution, may be lossless or lossy, meaning that the latter may lead to a quality degradation (Zinner et al., 2010). In addition, the transmission network may greatly affect the multimedia quality, namely due to major factors like packet loss, delay, and jitter (Minhas et al., 2012). Even with the adoption of a buffer at the receiver side, these network factors may cause the need for rebuffering, a streaming state activated when the playback buffer becomes empty and that leads to a playout stall, which is usually very annoying for the users (Pessemier et al., 2013;Ghadiyaram et al., 2014). Considering non-streaming services, the task completion (e.g., the non completion of a download) or the excessive amount of time a service takes to download or to upload are QoE degrading situations (Dellaert and Kahn, 1999), which are also caused by packet delay or data flow rate reduction. Other system factors that may affect the perceived quality are the type of device used at the users' side (e.g., the screen resolution, user interface capabilities, audio loudness, computation power, or battery lifetime) and some system specifications (e.g., interoperability, personalization, security, or privacy) (Ickin et al., 2012) the reader is suggested to refer to (Baraković and Skorin-Kapov, 2013;Siris et al., 2014;Liotou et al., 2015;Zhang et al., 2018) for more examples and details on QoE challenges concerning mobile networks.\n\n## 2.1.3. Context factors\n\nApart from the two aforementioned group of factors, there are external factors that influence the users' QoE by affecting the surrounding environment (Han et al., 2012). These context factors include temporal aspects, such as time of the day or day of the week (e.g., a better experience may be obtained when users are more relaxed, like during evenings or weekends), duration of the content and its popularity (e.g., users usually tolerate more distortion when they are watching popular videos), and service type, i.e., if it is live streaming or not (where users may have different quality expectations). The economic context can be also incorporated in this category of factors influencing QoE (Martinez et al., 2015), namely subscription type, costs, and brand of the system/service (including the availability of other service providers).\n\n## 2.2. QoE Estimation Methods\n\nMeasuring and ensuring good QoE in multimedia applications is very subjective in nature. Hence, one way to assess QoE is to perform subjective tests, which directly measure the perceived quality by inquiring persons about their opinion regarding the quality of the multimedia content that is being tested. The subjective test results can also be used to validate the objective assessment performance, which is another quality assessment methodology.\n\nThe Mean Opinion Score (MOS), which is standardized by ITU-T (ITU, 2016), is the most widely adopted QoE measurement. MOS is defined as a numeric value ranging from 1 to 5 (1-Bad, 2-Poor, 3-Fair, 4-Good, 5-Excellent) and it corresponds to the arithmetic mean of individual ratings in a panel of users. This approach has some drawbacks, namely it is costly, time consuming, and does not allow real-time evaluations. Moreover, some useful information may not be captured (e.g., if an impairment occurs at a certain moment but affects the overall QoE, this particular moment may not be detected).\n\nObjective quality methods have been developed in order to obtain a reliable QoE prediction while avoiding the need to perform subjective tests. The approach is based on mathematical techniques that yield quantitative measures of the multimedia content or service quality. Within the objective methods, two types of approaches can be identified: parameter-based methods and signalbased methods (Takahashi et al., 2008). The former rely on network/application parameters, such as viewing time (Balachandran et al., 2013), download ratio (Balachandran et al., 2014;Shafiq et al., 2014) or QoS parameters (Section 3.2 gives some examples of QoS-QoE mapping strategies). On the other hand, signal-based methods are based on the analysis of the signal; in intrusive methods, the analysis compares the received data with a reference, which can be the full original data (full reference methods) or some key features of it (reduced reference methods); non-intrusive methods, also known as no-reference methods, do not require access to the original multimedia content, relying only on the received signal to assess its quality. Nevertheless, some issues arise when performing objective assessment. Although intrusive methods are generally accurate, they are impracticable for monitoring live transmissions due to the need of the original multimedia content. Also, objective assessment may not reflect the perception of the users concerning the delivered service; for example, although some impairments may cause minor influence on the users' QoE, and therefore they could be disregarded, the same impairments may be detected and emphasized by the objective methods.\n\nIt is also important to mention that the majority of QoE estimation methods that have been proposed so far address the specific case of video quality evaluation, mainly due to the popularity achieved by video streaming services over the last years -surveys on video quality estimation can be found in (Chikkerur et al., 2011;Lin and Kuo, 2011), which mostly focus on objective quality models, and in (Chen et al., 2015c;Juluri et al., 2016), where the former work addresses metrics and methodologies relevant to the traditional video delivery, whereas the latter work focus on measurement mechanisms that are used to evaluate the QoE for online video streaming; a survey on parametric QoE estimation for popular services, such as video-ondemand streaming, Voice over IP (VoIP), web browsing, Skype and file download services, can be found in (Tsolkas et al., 2017).\n\nAs can be inferred from what was presented so far, QoE cannot be easily modeled and assessed due to the fact that its influence factors are very diverse and they may interrelate, as well as different users have different quality expectations. In order to attain an enhanced QoE evaluation, a combination of objective and subjective methods can be carried out (Rubino et al., 2006;Chen et al., 2010).\n\n## 3. Background on Scheduling Algorithms\n\nDistributing the available wireless resources among the users, i.e., multi-user scheduling, is one of the most important tasks that must be implemented in any wireless communication system. Specifically, a scheduler decides how users share the wireless channel by allocating radio resources such as power, time slots, frequency channels, or a combination of these resources. For instance, Time Division Multiple Access (TDMA) systems are characterized by having time slots as the radio resources units that can be assigned to a user; on the other hand, a scheduler allocates frequency channels in Frequency Division Multiple Access (FDMA) systems; in Orthogonal Frequency-Division Multiple Access (OFDMA) systems, radio resources are scheduled into the frequency/time domain -Fig. 3 depicts examples of resource allocation within TDMA, FDMA and OFDMA; for more detail about wireless multiple-access schemes, please refer to (Prasad and Mihovska, 2009;Molisch, 2011). Nevertheless, from a conceptual point of view, a scheduler can be designed in such a generic way that it is agnostic to which particular radio resources are handled by the underlying wireless multiple-access scheme -the scheduler only requires knowledge of the total amount of available resource units and the throughput provided by each of these units to each of the different users. As an example, suppose that each resource unit of Fig. 3, within its respective multiple-access scheme, yields the same throughput for all four users, and suppose also that the depicted time/frequency domain span corresponds to a single allocation decision. Accordingly, all three scheduling examples could be generated by the same generic scheduler, namely if the decision was to allocate 3/8, 2/8, 2/8 and 1/8 of the maximum achievable system throughput to user 1, to user 2, to user 3 and to user 4, respectively. For this reason, some of the proposed wireless resource scheduling techniques follow this generic approach and only point out the percentage of total resources that should be allocated to each user and who should be prioritized, leaving out which specific resources are being handled by the scheduler. Moreover, designing schedulers for wireless systems comprises many trade-offs among complexity, efficiency and fairness:\n\n• Complexity: It is important to limit the processing time of scheduling algorithms, since they usually have to perform their job under very short periods of time (e.g., 1 ms is the time that Long Term Evolution (LTE) schedulers have for allocation decisions (3GPP, 2019a)). In addition, scheduling schemes should be scalable, meaning that low-complexity algorithms should be preferred over very complex and non-linear solutions, which could be prohibitive in terms of computational cost, time, and memory usage when applied to scenarios with a large number of users.\n\n• Efficiency: Since radio resources are scarce, scheduling algorithms must aim at fully taking advantage of these resources. Performance indicators like the number of users served simultaneously or the average spec-tral efficiency of the wireless system are two examples of efficiency indicators adopted by many schedulers.\n\n• Fairness: A minimum performance must be also guaranteed for all users, in order to avoid unfair sharing of the wireless resources. Accordingly, implementing the fairness requirement in the scheduling schemes enables that users experiencing poor channel conditions (e.g., users that are far away from the base station) are also served, or that greedy users cannot provoke resource starvation in other users within the same wireless system.\n\nIn addition to the design factors described above, QoS and QoE provisioning must also be taken into account by the scheduling algorithms. In this section, some scheduling algorithms are reviewed, ranging from the simplest ones to QoS-aware approaches, followed by the introduction of QoS-QoE mapping strategies and utility-based optimization. QoE-based scheduling algorithms are presented in Section 4.\n\n## 3.1. QoE-unaware Schedulers\n\nAs previously mentioned, any scheduling strategy comprises many trade-offs among complexity, efficiency and fairness. In the case of schedulers that do not take QoE into account, these trade-offs also result from the significance that the different scheduling algorithms give to the communication channel characteristics and to QoS parameters.\n\n## 3.1.1. Channel-unaware Strategies\n\nThe schedulers that implement these approaches assume that the transmission channel is error-free and timeinvariant, which are unrealistic assumptions when dealing with wireless channels. Nevertheless, these strategies form the basis for more complex algorithms.\n\nFirst In, First Out (FIFO), also known as First Come, First Served (FCFS) (Arpaci-Dusseau and Arpaci-Dusseau, 2018), can be regarded as the simplest scheduling scheme, in which users are served according to the order of their resource request. Even though this approach is very easy to implement, it is not fair nor efficient.\n\nThe Round-Robin (RR) strategy (Arpaci-Dusseau and Arpaci-Dusseau, 2018) tries to add some fairness to the FIFO approach, namely by allocating an equal share of resources to each user in a round-robin manner. Thus, this scheduling algorithm is fair regarding the channel occupancy time of each user and can be considered the best choice if the transmitter does not know anything about the channel (Molisch, 2011). However, RR schedulers are unfair in terms of user throughput because they do not take into account the radio channel conditions (which have a major impact on the throughput).\n\n## 3.1.2. Channel-aware / QoS-unaware Strategies\n\nA wireless resource scheduler can take into account the channel state information that is usually fed back to the base stations, so as to enhance the efficiency of its scheduling algorithm.\n\nMaximum throughput (MT) (Prasad and Mihovska, 2009) is an example of a policy that, in each scheduling period, prioritizes the resources to the user experiencing the best channel conditions. Accordingly, these schedulers provide the highest system throughput, so that the best possible spectral efficiency is attained. Nevertheless, an MT scheduler is very unfair to users with poor channel conditions and can even make them suffer of starvation.\n\nThe concept adopted by Proportional Fair (PF) schedulers (Kelly, 1997) provides a compromise between fairness and spectral efficiency. Within this approach, the average throughput experienced in the past works as a weighting factor in an MT-like strategy, i.e., if two users can achieve the same throughput (taking into account the channel conditions), then the user that has experienced the lower average throughput is prioritized. This means that users with poor conditions will always be served after some time.\n\n## 3.1.3. Channel-aware / QoS-aware Strategies\n\nFor the purpose of attaining a certain performance level, different applications have different requirements, which are typically mapped into QoS parameters. Accordingly, scheduling algorithms should also take into account these QoS parameters. For instance, some scheduling strategies try to guarantee a minimum throughput for the users, whereas others deal with delay constrains. This last approach is more common among QoS-aware schedulers, since many applications, such as real-time flows, video streaming or VoIP calls, require that their packets are delivered within a certain deadline.\n\nThe Modified Largest Weighted Delay First (M-LWDF) algorithm (Andrews et al., 2001) and the Exponential/PF (EXP/PF) scheme (Rhee et al., 2003) are two of the most popular QoS-aware scheduling strategies, as they provide a balanced trade-off among fairness, spectral efficiency and QoS provisioning. Besides taking service delay requirements into account, both algorithms support different services and treat differently real-time data flows.\n\nAll the above mentioned scheduling strategies, either channel-unaware or channel-aware (with or without taking into account QoS requirements), are just an illustrative sample of what can be found in the literature. The reader is suggested to refer to the surveys by So-In et al. (2009); Afolabi et al. (2013); Asadi and Mancuso (2013); Capozzi et al. (2013);Abu-Ali et al. (2014); Castañeda et al. (2017) for more examples and details on scheduling algorithms that are not QoE-oriented, namely regarding WiMAX networks, multicast OFDMA systems, opportunistic scheduling, downlink in LTE networks, uplink in LTE and LTE-Advanced, and multi-user Multiple-Input Multiple-Output (MIMO) systems, respectively.\n\n## 3.2. QoS-QoE Mapping Strategies\n\nQoS-QoE mapping strategies have been presented to quantify QoE, thus making a transition from QoS-to QoEoriented optimization. QoS-QoE mapping relies on various QoS parameters, which can be divided into two levels: network QoS parameters (e.g., delay or packet loss rate) and application QoS parameters (e.g., rebuffering events or buffer level). Therefore, QoS-QoE mapping strategies try to discover the relationship between QoE and the two QoS levels, where the network QoS parameters are sometimes first mapped into application QoS parameters -cf. Fig. 4. Nevertheless, both types of QoS parameters can always be regarded as objective quality metrics, since their measurement is always well defined as they do not depend on any subjective judgment.\n\nChoosing a function φ : R → R that establishes a QoS-QoE mapping, i.e., a mapping between the objective quality metrics and a subjective score, is not a straightforward task. A linear mapping relationship could be adopted if a certain subjective quality difference always corresponded to the same proportional objective difference (Korhonen et al., 2012):\n\nwhere a and b represent the parameters determined by linear fitting the m th objective QoS metric versus the measured subjective scores. However, the perceived quality ratings usually do not present a linear behavior with respect to the practical objective quality metrics, meaning that linear mapping functions may lead to an inappropriate assessment of the performance. To overcome this issue, nonlinear mapping relationships have been adopted, discussed and compared (Korhonen et al., 2012;Alreshoodi and Woods, 2013); the most widely used can be summarized as follows:\n\n• Cubic polynomial (VQEG, 2010;Korhonen et al., 2012;ITU, 2015a,b):\n\n• Logistic functions (ITU, 2004;Korhonen et al., 2012;Song and Tjondronegoro, 2014):\n\n• Exponential function (Fiedler et al., 2010;Korhonen et al., 2012): • Power function (Korhonen et al., 2012):\n\n• Logarithmic function (ITU, 2014):\n\nConsidering the most general case, QoS metrics can be mapped into QoE values by performing a combination of the previous functions ( 1)-( 8), including intermediate nonlinear combinations of QoS metrics, i.e.,\n\nAs can be seen, QoE modeling through the use of QoS metrics may encompass complex relationships and interdependencies, with a parametrization that constitutes a non-trivial problem. Moreover, other issues arise when designing QoS-QoE mapping strategies, such as finding out which are the QoS metrics that are more useful for QoE prediction or how much data is needed to achieve a certain accuracy on the estimated QoE. Hence, and in order to tackle these challenges, some authors have proposed the use of machine learning techniques, with the final goal of devising complex models regarding the QoS-QoE relationship (Rubino et al., 2006;Balachandran et al., 2013Balachandran et al.,, 2014;;Shafiq et al., 2014;Yang et al., 2017;Casas et al., 2017).\n\nIn all cases, after computing the predicted QoE values, correlation analysis should be carried out between these and ground truth values, so as to assess the goodness of the mapping strategy. On the other hand, traditional QoS optimization techniques can be applied and assessed in QoE optimization scenarios by making use of QoS-QoE mapping strategies. For instance, Alfayly et al. (2012) investigated and evaluated the performance of three downlink schedulers (PF, M-LWDF and EXP/PF) in terms of QoE metric for VoIP applications over LTE, making it possible to choose the most suitable one in terms of subjective experience.\n\n$$QoE = φ 1 (QoS m ) = a + b • QoS m,(1)$$\n\n$$φ 2 (QoS m ) = a + b • QoS m + c • QoS 2 m + d • QoS 3 m.(2)$$\n\n$$φ 3 (QoS m ) = a + b 1 + c(QoS m + d) e,(3)$$\n\n$$φ 4 (QoS m ) = a + b 1 + exp[c(QoS m + d)],(4)$$\n\n$$φ 5 (QoS m ) = a + d 1 + (c • QoS m ) b e.(5)$$\n\n$$φ 6 (QoS m ) = a • exp(b • QoS m ) + c • exp(d • QoS m ).(6)$$\n\n$$φ 7 (QoS m ) = a • QoS b m + c.(7)$$\n\n$$φ 8 (QoS m ) = a • log(QoS m + b) + c. (8$$\n\n$$)$$\n\n$$QoE = i j m φ i [φ j (QoS m )].(9)$$\n\n## 3.3. Utility-based Optimization\n\nThe concept of utility functions emerged from microeconomics theory and formalizes the relationship between the service performance and the user perceived experience and satisfaction (Reichl et al., 2013). More specifically, the following utility function U : X → R relates all the resources a user could hypothetically have (set X) to real numbers, where U (x) > U (y) indicates that the user has a preference for x over y, with x, y ∈ X.\n\nThe utility-based scheduling optimization may then be regarded as a maximization of the total sum of users' utilities through Network Utility Maximization (NUM) techniques (Chiang et al., 2007). In mathematical terms, using NUM to allocate network resources (such as transmission power, time slots, etc.) corresponds to perform the following maximization:\n\nwhere U i corresponds to the utility function of the i th user, x i stands for the resources allocated to this user, and X max denotes the bounds of the available resources.\n\nIn many cases, scheduling can also be regarded as selecting a throughput vector R = [R 1,..., R K ], for all K users, from the current feasible throughput region R, i.e., the set of achievable throughputs expected for each user according to the respective allocated resources. Thus, the gradient-based scheduling algorithm (Stolyar, 2005) can be applied in order to perform resource allocation decisions:\n\nwhere U i denotes the derivative of an increasing concave utility function U i and R i stands for the achievable throughput expected for the i th user. For example, the MT scheduler can be obtained from ( 11) by adopting the utility function U i (R i ) = R i, whereas the PF policy derives from the gradient-based scheduling technique with a utility function\n\n, where R i represents the past average throughput experienced by the i th user; accordingly, the selected MT and PF throughput vectors, R * M T and R * P F respectively, are given by\n\n$$maximize i U i (x i ) subject to i x i ≤ X max,(10)$$\n\n$$R * = arg max R∈R i U i (R i ) • R i,(11)$$\n\n$$U i (R i ) = log(R i )$$\n\n$$R * M T = arg max R∈R i R i,(12)$$\n\n$$R * P F = arg max R∈R i R i R i.(13)$$\n\n## 3.4. Discussion\n\nBased on what was previously described, QoS-QoE mapping strategies and utility-based optimization can be regarded as the fundamental tools to perform the shift from QoS-to QoE-oriented scheduling. Ideally, one should aim at obtaining mathematical formulas that relate application, transport, and physical layer parameters to subjective quality experienced by the users. With these, wireless systems design can be adjusted in order to improve the quality perceived by the users. For instance, a closedform expression was presented by Colonnese et al. (2016) concerning the probability of timely transmission of video sequences as a function of the users' allocated bandwidth. As a consequence, and given a certain QoE requirement based on the probability of timely delivery and the received video stream quality level, the aforementioned expression allows to infer the number of users that can be accommodated in the wireless access system, as well as it can be used to design admission procedures, bandwidth pricing policies, and cell dimensioning. In (Hoßfeld et al., 2017), a general fairness metric is formulated for shared systems, which satisfies QoE-relevant properties, assuming that estimated QoE values are known. This QoE fairness metric may be adopted when comparing different resource management techniques in terms of their fairness across users and services, although it says nothing about how good the system is and thus needs to be considered together with the achieved (e.g., mean) QoE in system design.\n\nIn certain cases, the mathematical formula adopted for a QoS-QoE mapping strategy can also be used as a QoEoriented utility function, namely if there is a well-defined relation between allocation decisions and the considered QoS parameters. For instance, if a QoS-QoE mapping formula regarding video streaming considers, as single input, the transmitted video bitrate, and assuming that this QoS parameter is directly proportional to the achievable throughput, then a utility function can be derived from this QoS-QoE mapping formula, namely by replacing the QoS input by the corresponding relation between transmitted video bitrate and achievable throughput. Another example of a utility function that stems from QoS-QoE mapping strategies can be given regarding file download applications, namely when a QoS-QoE mapping formula only considers, as QoS input, the service response time (i.e., the file download time), which is inversely proportional to the achievable throughput, with a constant of proportionality equal to the size of the file that is being downloaded.\n\nOn the other hand, in many cases, the inputs of QoS-QoE mapping strategies might not have a well-defined relation with the allocation decisions (e.g., when packet loss rate is adopted as QoS input). Nonetheless, utility functions (or, alternatively, throughput vector selection formulas) can be designed without any knowledge of a specific QoS-QoE mapping strategy and still follow a QoE-oriented approach, as long as the scheduling goals include addressing some issues that affect the users' QoE -for instance, to try to lessen the impact of packet losses by serving better the respective users afterwards; another example is to perform allocation of resources in order to try to avoid rebuffering events. Accordingly, very often QoS-QoE mapping strategies are only required to assess the goodness of a utility function/throughput vector selection formula, i.e., to know the impact of a certain resource allocation policy on the users' QoE. Notice that this last approach can be used to study, in terms of QoE, any scheduling procedure, even those that do not follow a QoE-based design. For example, and considering video streaming over LTE, the QoE metrics presented in (Yaacoub and Dawy, 2014) try to describe the performance of radio resource management methods regarding the end-users subjective video quality. The authors measured minimum, average and geometric mean QoE when scheduling algorithms like MT and PF are adopted. In (Abbas et al., 2016), the QoE of adaptive video streaming is analyzed under several scheduling policies, like RR and MT, where the QoE is based on the mean video bit rate and the mean buffer surplus. The examination of the performance impact of the different scheduling schemes is then used to suggest the best strategy to be adopted in various mobility scenarios.\n\n## 4. QoE-oriented Scheduling Algorithms\n\nMany challenges arise when attempting to perform QoEoriented wireless resources scheduling. As seen in the previous sections, it is important to identify the factors that influence QoE and their relationships to QoE metrics for a given type of service. Some more challenges may be identified after addressing the QoE modeling, namely determining which parameters to collect (e.g., user requirements, network performance, application type, context, etc.), where, how, and when to collect them (e.g., the required parameters could be collected at the base stations or at the end-user devices, either before, during, or after the delivery of the service). Lastly, procedures have also to be defined in order to combine all these steps, i.e., it is necessary to design the methods that allow the collected data to perform QoE-aware scheduling.\n\nIn this section, state-of-the-art QoE-based scheduling strategies for wireless systems are reviewed, highlighting the parameters adopted for QoE optimization. To simplify the reading of the survey, the strategies that address the downlink scenario have been classified into three categories -cf. Fig. 5: (i) passive end-user device; (ii) active end-user device / passive user; (iii) active end-user device / active user. This classification is based on the adjustments required, at the end-user devices, in order to implement the different scheduling strategies on wireless systems. The last part of this section provides a review of QoE-aware scheduling methods that can enhance the wireless resources management in other scenarios, namely the uplink direction, the multi-cell case, under heterogeneous, cognitive radio, relay and multi-user MIMO networks, as well as when dealing with energy-related issues.\n\n## 4.1. Passive End-user Device Strategies\n\nScheduling techniques are easier to implement in a wireless network when the users, as well as their devices, do not perform any exclusive QoE tasks (e.g., monitoring, measuring and reporting relevant parameters), as the QoE assessment is based on measurements that can be carried out solely at the base station side. Since the required assessments can be performed by the scheduler on the network side, no extra information needs to be exchanged between the user's device and the network. On the other hand, these approaches may not achieve the best possible QoE performance, since many relevant metrics, which could be collected at the end-user device (e.g., buffer status), either cannot be used or have to be estimated.\n\n## 4.1.1. Video Streaming\n\nThe simplest QoE-based scheduling approaches consider only the impact of the throughput on the user-perceived quality, namely by adopting the following utility function:\n\nWith respect to video streaming, Shehada et al. (2011) and Thakolsri et al. (2011) made use of ( 14) by first establishing a mapping between video bitrate and MOS, followed by an allocation of resources to each user assuming that the bitrate of the transmitted video is adjusted to match the respective achievable throughput, so that there is a known correspondence between throughput and MOS. Besides considering the NUM, it is also proposed in (Shehada et al., 2011) another allocation criterion that establishes an a priori target mean MOS of all users, in order to save some network resources (which could be used to serve more users or to support high-demand applications), whereas a tuning mechanism is presented in (Thakolsri et al., 2011) that enables the network operator to dynamically adjust the resource allocation between similar perceived quality among all users (system fairness) and maximum average perceived quality (system efficiency). Yu et al. (2018) proposed a framework to optimize the throughput distribution, which also comprises the determination of video encoding parameters for each user, so that the combined video compression plus radio resource allocation is able to maximize the QoE of all users. Nevertheless, the previous works neglect the impact of packet loss on the QoE, a relevant parameter that was taken into account by other authors. For instance, a trained random neural network is used in (Piamrat et al., 2010) to establish a mapping between the packet loss rate as well as the mean loss burst size and a video QoE score normalized to scale [0, 1]; next, the respective score of each user, e i, is adopted as a coefficient in modified versions of the MT and PF algorithms, i.e.,\n\nwhere w i corresponds to the respective i th user weight associated to the MT scheduling policy (w i = 1) or the PF one (w i = 1 Ri(n) ). In (Ju et al., 2012), throughput was also considered in conjunction with packet loss rate in the work presented, in which an artificial neural network is adopted to learn the relationship between these parameters and the QoE; afterwards, the scheduler allocates resources based on a particle swarm optimization method, which has the goal of maximizing the users' QoE and, at the same time, balance fairness among them. Ai et al. (2012) also made use of the utility function ( 14), but now throughput is replaced by goodput (i.e., the rate at which the useful data -namely excluding retransmitted data packets -is delivered), so packet loss rate can be considered implicitly, as well as the resource allocation algorithm proposed therein also aims at decreasing the video quality variation.\n\nThe aforementioned scheduling techniques do not take into account the occurrence of playout stalls (which, as mentioned in Section 2.1, are very annoying for the users), mainly because these algorithms assume that the bitrate of the transmitted video is adjusted to match the respective achievable throughput -hence, in theory, rebuffering events would be avoided. However, clients may request video segments with specific bitrates, which means that not only the allocation algorithms must be able to take bitrate constraints as input parameters, but also they should aim at providing interruption-free video transmissions, as playout stalls are more likely to occur if the requested bitrate by a client is too demanding when compared to the respective achievable throughput. One way to tackle this problem is to consider that the radio resource assignment is given by\n\nwhere γ i stands for a term that reflects the effect of the i th user satisfaction based on the possibility of rebuffering events taking place. For instance, Wirth et al. (2012) made use of ( 16) by defining γ i as an exponential weighted moving average filter that depends on the minimum throughput requirement R min i (which stems from the requested video bitrate):\n\nwhere α * i stands for the α i value of the previous scheduling period and denotes a small positive value. With this approach, a user will be prioritized if the respective achievable throughput does not meet the minimum rate constraint (in order to try to avoid playout stalls), whereas if this target is met, then the user experiences a very low γ i weight, thus being deferred from being served. Another approach based on ( 16) is proposed in (Seyedebrahimi et al., 2014), namely by considering w i = 1 and a γ i weight given by\n\nwhere β corresponds to a value that can be adjusted to achieve a certain trade-off between fairness (high β values) and efficiency (β close to zero).\n\nStill regarding the occurrence of playout stalls, Pastushok and Turlikov (2016) proposed a lower bound for a mean rebuffering percentage (percentage of the entire streaming time in which a user is experiencing playout stalls), along with a corresponding optimal scheduling strategy. Their approach yields, for each user, an achievable throughput value that ranges from zero to R i = R min i -more precisely, the scheduler serves the users that require less resources (in order to fulfill their R min i demand) until there are no more resources available, thus meaning that users with a high R min i Ci relation are more prone to be left out, where C i represents the maximum achievable throughput if all resources were assigned to user i. Nevertheless, this scheduling technique is not suitable for the case where all users can be served at the same time without network congestion, namely because some spare resources would not be allocated (which could lead to an increase of the initial playout delay). In addition, the aforementioned lower bound assumes that the video representation quality chosen at the beginning will be the same throughout the whole streaming session, which is not true if adaptive video streaming is enforced.\n\nAnother factor that may influence the users' QoE, and which has not been addressed by the previous scheduling algorithms, is when one or more subscribers should be prioritized over the remaining because, e.g., they are paying more in order to obtain a better service. One way to tackle this issue is to divide the users into different classes and assign different priority weights to them. This approach was followed by Hsieh and Hou (2018), in conjunction with a scheduling technique that tries to minimize the duration of playout stalls. More specifically, it is proposed to schedule the client with the largest R i in each scheduling period and, if a tie occurs, the chosen client is the one that has the smallest\n\n, where ν i corresponds to a predetermined weight that takes into account the user's class. In order to compute ν i, the authors adopt the NUM technique and obtain some tractable solutions; however, it is important to mention that the scheduling technique presented in (Hsieh and Hou, 2018) is designed assuming some conditions, namely that the sum of the minimum throughputs required by the users is not higher than the maximum achievable system throughput. Moreover, since a user will always be sacrificed if the respective channel conditions are poorer than the ones of another user, this scheduling policy is more suitable in scenarios where the throughput of the wireless link is expected to be similar among all users.\n\nSome authors have also considered another relevant parameter not addressed so far, namely the Head-of-Line (HoL) packet delay (i.e., the delay of the first packet), in order to perform QoE-oriented scheduling that is somewhat capable of minimizing playout stalls. In (Chandur and Sivalingam, 2014), a modified version of the PF algorithm is proposed, which adopts the following scheduling decision:\n\nwhere τ i and τ i denote the HoL packet delay and the average packet delay, respectively, regarding the i th user, whereas a and b represent some constants which enable to adjust the impact of the delay variables on the scheduling decision. An approach based on the M-LWDF technique is proposed in (Li et al., 2016), in which the scheduling process jointly considers packet delay, expected throughput and video importance. More specifically, before scheduling a user, some overdue packets are discarded within this scheme, namely those with an associated delay that has exceeded the deadline threshold given by p (2) i / p (1)\n\n$$U i (R i ) = f (R i ), f : R → M OS. (14$$\n\n$$)$$\n\n$$R * = arg max R∈R i (2 -e i ) • w i • R i,(15)$$\n\n$$R * = arg max R∈R i γ i • w i • R i,(16)$$\n\n$$γ i = e α * i α i α,(17)$$\n\n$$α i = max R min i -R i,,(18)$$\n\n$$γ i = 1 - R i R min i β,(19)$$\n\n$$ν i R i -R min i$$\n\n$$R * = arg max R∈R i R i R i + a • exp{b(τ i -τ i )} R min i R i,(20)$$\n\n## i\n\n, where\n\ni and p\n\n(2)\n\ni correspond to positive tuning parameters that enable to adjust the deadline threshold with respect to the i th user. Afterwards, radio resource assignment is performed by following the scheduling rule given by\n\nwhere I i stands for the video importance index of the packet that is being download by the i th user, which is derived from an algorithm described in (Li et al., 2016) and aims at setting a value for how important is a packet in enhancing the video quality. Khan and Martini (2016) proposed a scheduling policy that is similar to the previous one, in the sense that not only it takes into account packet delay, expected throughput and video importance, but also the scheduler is allowed to discard some packets. However, the packet filtering process is now based on the respective contribution towards video quality (instead of overdue packets), under an algorithm that is adjusted to work with scalable video streaming. The scheduling decision of this method is as follows:\n\nwhere P i corresponds to the priority rating of the packet that is being downloaded by the i th user, which takes into account the respective bitrate and contribution towards the perceived video quality, and q i denotes the number of packets currently residing in the queue to be scheduled to user i. It is noteworthy to stress that these last two scheduling techniques are not lossless, i.e., a client may not receive all the information it asked for; on the other hand, resources can be saved (and further allocated in order to enhance the QoE) by not transmitting overdue packets or with low contribution to the video quality, which might have become useless at the receiver in the sense that they would not provide a great QoE enhancement.\n\n$$p (1)$$\n\n$$R * = arg max R∈R i p (1) i • I i • τ i • R i,(21)$$\n\n$$R * = arg max R∈R i exp{[P i ] τ • τ i } R i R i q i,(22)$$\n\n## 4.1.2. Other Applications\n\nThe user experience in some wireless applications other than video streaming can also be enhanced by the use of QoE-aware scheduling methods. With respect to VoIP, the algorithm presented in (Chen et al., 2015a) tries to allocate resources in order to limit the delay within a certain deadline and, consequently, meet the tight delay requirements of VoIP; in addition, the proposed framework also has the goal of minimizing the total number of radio resources scheduled during a certain period of time, a procedure which is based on not serving some users at some time instances if their forecasted channel conditions are able to cope with future transmissions that still meet the deadline. Ameigeiras et al. (2010) addressed web browsing applications and suggested a mapping from user throughput to user experienced quality, which enables to perform radio resource allocation through the maximization of the aggregate utility over all users; the respective utility function is given by\n\nwhere W S stands for the Web page size -note that this approach can also be regarded as one that aims at rendering the Web page with a service latency lower than approximately 10 seconds, namely because the ratio W S Ri is intended to correspond to the service response time measured in seconds.\n\nIn the general case, wireless networks provide multiservices, where video streaming, VoIP, Web browsing and file download applications are available to the users; consequently, there should be a scheduling concern of providing high QoE for all of them. For instance, the scheduler proposed in (Liotou et al., 2016) tries to maintain the past average throughput values per user, in order to moderate throughput fluctuations; more specifically, the authors adopt the following scheduling rule:\n\nNoticing that this approach is application-unaware, it has the advantage that there is no need for the scheduler to know which service each user is using. On the other hand, and since QoE is also application-dependent (as mentioned in Section 2.1), scheduling strategies should be adjusted in order to take into account the particularities of each service. For instance, the work presented in (Liu et al., 2012) proposes a NUM-based scheduling scheme, in which different utilities functions are adopted for each servicethe authors consider mapping functions based on throughput plus packet loss regarding video streaming, the delay is regarded as the relevant parameter for VoIP, whereas the QoE of Web browsing and file download applications are based on service response time and throughput, respectively. This scheduling algorithm was assessed for two operation modes, namely one that aims at maximizing the sum of all users' QoE and another with the optimization target of maximizing the sum of the logarithm of the users' QoE -it is claimed that the latter mode improves the fairness among services without a great impact on the average QoE. The resource allocation scheme described in (Wang et al., 2017) also aims at maximizing the average QoE regarding multi-services; the authors devised a personalized strategy, in which QoE is evaluated not only using QoS factors, such as throughput, packet loss rate or delay, but also considering a predicted user preference based on contextual factors (e.g., the registered age, gender and occupation of the user, time of the day, day of the week, duration of the content and its popularity, etc.). Anand and de Veciana (2017) designed a scheduling method which takes into consideration the mean flow delays in multiservice systems. More specifically, they propose a framework based on the Gittins index to solve the optimization problem given by inf\n\nwhere the d π = [d π 1,..., d π S ] denotes the mean delay vector realized by the scheduling policy π (for all S services) from the feasible delay region D, i.e., the set of possible mean delay vectors considering all policies; with respect to service s, λ s stands for the arrival rate, d s represents the mean delay experienced and f s (•) corresponds to the cost function that reflects the respective QoE sensitivity.\n\nIn (Sacchi et al., 2011), a wireless resources assignment method is presented that also has the goal of providing a similar QoE among all users (which have miscellaneous requirements for video, voice and data services), namely by making use of game theory concepts, as well as minimum throughput requirement and packet loss probability of the different services, in order to maximize the minimum QoE. A scheduling procedure is proposed in (Xin et al., 2014) with respect to instant messaging service, which can incorporate video chat, audio chat and text chat subservices, and where a user may launch several subservices at the same time (e.g., multiple text chats and an audio chat). More precisely, the authors adopt the average delay of image, voice and text flows as the base metric to quantify QoE, along with a method in which the scheduler computes, without feedback from the terminals, the probability that a user is focusing on a certain subservice (taking into account the subservice type and its serving quality), thus regarding the one with the higher probability as the representative service. Based on this premise, the scheduling scheme first looks up for the user experiencing the poorest QoE of the representative service and then allocates resources for one subservice of this user; regarding this last step, a random approach is devised, such that even though the representative service has a higher chance of being prioritized, the other subservices will not be starved (which also helps to handle inaccurate estimations of the representative service).\n\nIt is important to stress that even though the previous approaches aim at ensuring that all users have an identical QoE (system fairness), this might lead to undesirable situations: for instance, when one user is requesting a very demanding service or is experiencing very poor channel conditions, a QoE-fair scheduler would allocate more resources to this user and eventually force the majority of the remaining users to have a poor experience. On the other hand, a scheduler that maximizes the average perceived quality (system efficiency) could also sacrifice some users by not providing them, at least, an acceptable QoE. For this reason, and regarding multi-service systems, some works proposed solutions that try to offer a trade-off between system fairness and system efficiency (Deng et al., 2014;Fei et al., 2015;Monteiro et al., 2015;Rugelj et al., 2014;Hori and Ohtsuki, 2016) -although these works are somewhat similar, their differences deserve to be high-lighted, which will be done in the following paragraph.\n\nAccording to the scheme introduced in (Deng et al., 2014), the users are served following an MT fashion until all of them have the respective minimum throughput requirement satisfied (it is assumed that QoE is estimated by taking into account solely the throughput of video, audio and file download applications); in the following step, the resources are allocated to the users that can achieve the best QoE gain. In (Fei et al., 2015), the proposed scheduler performs the same first step as the previous solution, i.e., users are served following an MT fashion until all of them have the respective minimum throughput requirement satisfied; afterwards, the remaining resources are allocated in a fair manner such that almost the same quantity is assigned to all users. In (Monteiro et al., 2015), the proposed algorithm serves the users following an MT fashion until a predefined number of users is satisfied by having a QoE equal or greater than a certain threshold (it is assumed that throughput can be mapped into QoE); afterwards, the remaining resources are allocated to the users with the lowest QoE. Rugelj et al. (2014) presented a method that searches for the users with the minimum QoE and assigns resources to them; this process is repeated until each user is considered satisfied according to the respective minimum QoE (the authors adopted mapping functions based on throughput plus packet loss regarding video streaming and audio applications, whereas for Web browsing applications the considered relevant parameter was service response time); next, the remaining resources are allocated following an MT fashion, with the proviso that users that achieve a very high satisfaction threshold are excluded from being further served. The approach of Hori and Ohtsuki (2016) is very similar to the previous one, with the main difference that all users are considered satisfied according to the same QoE threshold (the adopted mapping functions are also slightly different, namely delay is now used as relevant parameters for audio applications); this criterion might be less realistic than the previous one, in the sense that users often expect a different level of satisfaction for different services.\n\nMore recently, El-Azouzi et al. ( 2019) proposed a scheduling strategy that prioritizes video streaming over other applications; in particular, resources are allocated to video users according to the policy\n\nhowever, if the average throughput of all the video flows is above their respective throughput requirement (i.e., ∀ i : R i > R min i ), then the spare resources are assigned to non-video traffic, thus improving the system utilization. The work of Chandrasekhar et al. (2019) also treats video delivery in a special way, namely by using real-time network-based machine learning classifiers, which make use of standard unencrypted packet headers in order to not only detect the service type of different flows, but also to estimate the player status regarding video streaming users; the authors claim that their framework is able to provide an enhanced video QoE (with an acceptable impact on other non-video services), namely by increasing the prioritization weight of users detected as being experiencing a rebuffering event.\n\n$$U i (R i ) = 5 - 578 1 + 11.77 + 22.61 • Ri W S 2,(23)$$\n\n$$R * = arg max R∈R i 1 R i -R i. (24$$\n\n$$)$$\n\n$$d π s λ s f s (d s ) | d π ∈ D,(25)$$\n\n$$R * = arg max R∈R i (R min i -R i ) • R i ; (26)$$\n\n## 4.1.3. General Considerations on Passive End-user Device Strategies\n\nThe previous overview demonstrates that the research community has made some efforts to present scheduling methodologies that enhance the QoE of the users and which rely on measurements that can be carried out solely at the base station side -these strategies are summarized in Table 2. In accordance with the popularity achieved by video streaming services over the last years, the majority of the works proposed QoE-oriented scheduling solutions taking into account this type of service. Moreover, many of these works aim at allocating resources with the goal of proving interruption-free video; however, their approach may still be inefficient: for instance, consider a user that, at a certain scheduling period, has a large amount of data stored in the respective buffer -e.g., the user had paused the playback for a while and, in the meantime, the buffer stored some dozens of seconds of video; if, by following the previously mentioned algorithms, this user is prioritized, then this scheduling decision occurs in a situation where the respective video could be played smoothly during some time, even if this user was not served. Accordingly, this user is being favored over others that might have nearlyempty buffers and, if not served, will experience playout stalls.\n\nMany proposals also addressed the multi-service case, in which the adopted objective functions have a dual role: not only they have to capture the particularities of each service regarding QoE by using relevant metrics that enable to accomplish that goal (e.g., throughput plays a central role when designing QoE-oriented schedulers for the video streaming service, whereas delay is of prime importance for VoIP applications), but also the different objective functions establish a trade-off regarding the different services, namely which service should be prioritized and when this should occur. For instance, users that are performing web browsing or large file downloads may not perceive a QoE degradation if the respective service delay is slightly increased; hence, considering a congested system, it may be beneficial to prioritize applications such as video streaming or VoIP, so that their QoE is not affected or is even increased. On the other hand, considering uncongested systems, scheduling more resources to the latter applications would enhance their QoE only marginally if these applications are already properly served; in this situation, it would make more sense to allocate the remaining resources to web browsing or large file download applications. Therefore, the choice of objective functions that are able to meet simultaneously the two aforementioned goals turns multi-service QoE-oriented scheduling in a tougher challenge when compared to single-service systems. In addition, there is also another issue that usually arises within multi-service systems, namely the need for the scheduler to know which service each user is using, a requirement which either involves detecting the service type through packet inspection or entails an extra cooperation between the application-level server and the scheduler.\n\n## 4.2. Active End-user Device / Passive User Strategies\n\nOne way to enhance the subjective quality perception of a service is to perform QoE assessment as close as possible to the end-user and to report this information to the base station. In this way, there is a higher degree of confidence regarding the influence of the scheduler adjustments versus the QoE improvement, since more accurate QoE metrics can be used. However, this type of scheduling algorithms require feedback channels for QoE measurements reporting. For instance, the Dynamic Adaptive Streaming over HyperText Transfer Protocol (DASH) technolas MPEG-DASH (ISO, 2014), and its extensions standardized by the 3 rd Generation Partnership Project (3GPP) for DASH use over wireless networks (3GP-DASH) (3GPP, 2019b) already incorporate specifications about how relevant QoE parameters (e.g., video buffer level) can be reported to the server -it is noteworthy to mention that one of the goals of this technology is to cope with variable network conditions, such as those caused by wireless link quality variations, namely by defining how different representations (with different bitrates) of the same multimedia content can be split into smaller segments, which can be independently decoded by a client, and how the clients can select and retrieve these segments from a DASH server; accordingly, a DASH client is able to switch seamlessly between different representations during a streaming session, an adaptation which, for example, enables to minimize the impact of throughput fluctuations on playout stalls, thus enhancing the QoE.\n\nIn the scheduling strategies described in this subsection, the end-users devices are regarded as clients that are able to measure and to report to the base stations the QoE metrics that will be be taken into account by the scheduler, while the user itself does not perform any action. (b) Scheduling procedure.\n\nFigure 6: Flowchart of an active end-user device / passive user scheduling algorithm for video streaming that runs on top of traditional schedulers (Wamser et al., 2012;Pervez and Raheel, 2015).\n\n## 4.2.1. Video Streaming\n\nThe buffered playout time (at the end-users devices) of YouTube videos is considered in (Wamser et al., 2012;Pervez and Raheel, 2015) in order to generate signaling events when the buffered video playout time drops below threshold t b or goes above threshold t a (t a ≥ t b ), cf. Fig. 6: in the former case, video flows are tagged as being in a critical state, thus the scheduler prioritizes the respective packets; when the buffered playout time of a critical video flow becomes greater than t a, it is relabeled as a normal flow and the scheduler allocates resources according to any scheduling policy (e.g., MT, PF, etc.). This methodology, which is intended to run on top of traditional scheduling algorithms, is not proactive regarding overall QoE improvement, i.e., instead of aiming at QoE amelioration for all users at all scheduling instances, this scheme only prioritizes video flows when QoE impairment, namely a playout stall, might be imminent as a result of a buffer that is becoming empty. Another scheduling method that attempts to avoid QoE degradation is presented in (Seetharam et al., 2015), in which the goal is to maximize the minimum buffered video playout time among all users. On the other hand, a more proactive approach can be adopted, in order to also try to enhance the users' overall QoE. For instance, Liu et al. (2015) proposed a scheduler that considers QoE ∝ 1 D M OS, where D M OS is a positive value that denotes the overall perceived quality deterioration, which is not only based on ongoing rebuffering events, but also takes into account past playout stalls; accordingly, and in order to provide a fair QoE, the user with the highest D M OS is prioritized in each scheduling period -still regarding the fairness issue, a protection mechanism is also conceived so that a user is not continuously prioritized over the others for more than a certain amount of time.\n\nIn general, the aforementioned scheduling techniques do not take advantage of the higher throughputs that can be attained by the users that are experiencing better channel conditions -if these users are better served, then the average perceived quality (system efficiency) may be improved, provided that the users that are experiencing poor chan-nel conditions can still achieve a satisfactory QoE. Hence, some scheduling procedures have been devised that make use of the reported buffer level and aim at increasing the system efficiency regarding QoE. In (Joseph and de Veciana, 2014), the following scheduling rule is proposed for radio resource assignment:\n\nwhere v i stands for an indicator of risk of violation of rebuffering constrains by the i th user and\n\nfor some constant v (typically set as zero), and is strictly increasing for v i ≥ v -the authors suggest an algorithm in which the parameter v i is updated, in each scheduling period, symmetrically with respect to the buffered playout time of user i, denoted from now on as B i, i.e., v i is roughly a linear decreasing function of B i. Some proposals have also been presented which are based on the PF technique, in the sense that they follow the PF scheduling behavior but, whenever the buffer level of some users starts running low, they also carry out a smooth behavior transition in order to allocate more resources to these users. For instance, the scheduling technique introduced in (Navarro-Ortiz et al., 2013) -which was also adopted for multicast systems by Yuan et al. (2017) -aims at prioritizing users with low buffer levels, with the proviso that these users will not consume an excessive amount of resources; this goal is pursued by the scheduling rule\n\nwhere L(•) denotes a decreasing logistic function -hence L(B i ) is upper bounded for low B i, thus avoiding a monopolization of the resources by one or more users with low buffer levels. In (Ramamurthi and Oyman, 2014), the variation rate of the buffer level is also taken into account when performing resource allocation, with the purpose of a continuous adjustment of the scheduling priorities in order to prevent low buffer levels in the first place; the corresponding resource allocation decision is as follows:\n\nwhere η i represents a parameter that determines the timescale over which rebuffering constrains are enforced for the i th user -this parameter reflects the current buffer level, in the sense that the authors suggest an algorithm that scales η i to prioritize users with low buffer levels -and T i corresponds to a video-aware user token parameter, which should increase or decrease whenever the variation rate of the buffer level is below or above a certain threshold, respectively. The strategy proposed in (Singh et al., 2012) also tries to provide fairness in terms of rebuffering percentage (i.e., the percentage of the total streaming time spent rebuffering); accordingly, the respective scheduling decision is given by\n\nwhere F i corresponds to the size of the video frame that the i th user is downloading, f i stands for the number of video frames in the buffer of this user, and f min, which is tunable, represents the minimum number of frames that each client should have in the buffer; the variable p rebufi denotes the rebuffering percentage of the i th user, N U corresponds to the total number of connected users, whereas µ and ξ stand for other tunable parameters of this scheduling algorithm.\n\nIt is noteworthy to mention that even though the previous scheduling strategies exploit the higher throughputs that can be attained by some users, this approach might still be inefficient -for instance, prioritizing the user that is experiencing the highest throughput may not imply that the respective buffer level will have a great increment (e.g., this user could be requesting a very high quality video); therefore, it might be more efficient to serve other users which can store a higher amount of playout time in their buffers, in order to better obviate the occurrence of stalls. This approach is followed by the scheduling algorithm introduced in (Rodrigues et al., 2018), namely by considering explicitly the predicted buffer level variation of the user i, ∆B i, according to the resources that could be allocated to this user:\n\nwhere Ω denotes an emergency state threshold (somewhat equivalent to the critical state of Fig. 6), which is activated when one or more users have a buffered playout time lower than Ω, i.e., this emergency condition is useful in preventing some users from suffering from starvation (a situation which would give rise to playout stalls). Some authors also presented scheduling techniques that follow a methodology that is different from what was seen so far; more precisely, the algorithms proposed in (Pu et al., 2012;Essaili et al., 2015;Zhao et al., 2015;Li et al., 2017;Kumar et al., 2017) adopt a proxy-based approach, in which not only users with low buffer levels are prioritized, but also the proxy is able to modify the request of a client and serve a video segment with a lower bitrate, namely if this last procedure is able to avoid impending playout stalls. In spite of the fact that the use of this type of proxy is very helpful in maximizing considerably the number of interruption-free transmissions, there are also some issues that arise and that must be considered before implementing it: first, the video streaming system needs to provide a seamless switching between different bitrate representations, of the same multimedia content, during a streaming session (e.g., a DASH-based approach could be followed); secondly, a client may not be willing to accept a bitrate representation which is different from the one it requested (for instance, although the DASH specifications foresee the case where alternative representations could be admissible at the client side, this option can only be used if the client activates it first).\n\n$$R * = arg max R∈R i h v i (v i ) • R i,(27)$$\n\n$$h v i (•) denotes a non-negative Lipschitz continuous function such that lim v→∞ h v i (v) = ∞, h v i (v i ) = 0 for all v i ≤ v$$\n\n$$R * = arg max R∈R i R i R i L(B i ), (28$$\n\n$$)$$\n\n$$R * = arg max R∈R i R i R i exp {η i • T i }, (29$$\n\n$$)$$\n\n$$R * = arg max R∈R i R i R i + µ • R i F i exp {ξ(f min -f i )} V i,(30)$$\n\n$$V i =    1 + N U ×p rebuf,i N U j=1 p rebuf,i, if N U j=1 p rebuf,i > 0 1, otherwise,(31)$$\n\n$$R * =        arg max R∈R i 1 B i, if ∃j : B j < Ω arg max R∈R i ∆B i B i, otherwise,(32)$$\n\n## 4.2.2. Other Applications\n\nTurning the attention to applications other than video streaming, a wireless resources redistribution algorithm is proposed in (Szabó et al., 2016) for web browsing, where the page state information is adopted as one of the inputs of the scheduler -more specifically, the proposed method subtracts some capacity from a client when the respective web page enters the \"interactive\" state, i.e., when something first renders on the screen and the respective user can start browsing; afterwards, the liberated resources are redistributed to those clients that have web pages in the \"loading\" state, i.e., the initial downloading part where the respective users are awaiting something to be displayed and which corresponds to the most sensitive period in terms of QoE. Nguyen et al. (2017) presented a scheduling strategy for real-time services (especially for voice, but it can also be used for video flows), which follows a PF-like approach with the addition of a multiplying term that takes into account a QoE metric computed at the devices and reported to the scheduler (this QoE metric depends on the delay, packet loss rate and network jitter); thus, the users that have higher a QoE have a higher probability of being prioritized under this scheduling algorithm, which could pose some difficulties in terms of fairness.\n\n## 4.2.3. General Considerations on Active End-user Device\n\n/ Passive User Strategies As can be inferred from the QoE-oriented scheduling solutions overview performed in this subsection, which are summarized in Table 3, the possibility of reporting relevant QoE metrics, from the users' devices to a scheduler, is mainly useful for video streaming services. More specifically, the report of information that is only available at the device, such as the buffer level, enables to design scheduling algorithms that are more capable of attaining the interruption-free video goal, thus providing an enhanced QoE when compared to passive end-user device techniques. Moreover, it is noteworthy to point out the usefulness of using real QoE metrics instead of estimated ones: for instance, even though a scheduler could try to estimate the users' buffer level at the network side (i.e., without any buffer level report from the end-user devices to the network, which would avoid the need for feedback channels), several factors may sometimes turn this estimation task into a meaningless one -e.g., different initial playout delays (an information which is usually not known by the network) lead to different estimated buffer levels, as well as when a user pauses a video but the respective video download is not interrupted.\n\nOn the other hand, the reported QoE metrics should be incorporated carefully by the different scheduling methods, i.e., these should take into account the different states of each service and should not regard the same information always in the same way, so that the final QoE is optimized -for instance, consider a scheduler that prioritizes the users that have a low buffered playout time; noticing that a playout stall is usually more annoying than a slightly longer initial playout delay, this scheduler should refrain from giving a high priority to those users that are starting a video download (although their buffer level is closer to zero), because otherwise this could entail a lack of resources for already existing flows.\n\n## 4.3. Active End-user Device / Active User Strategies\n\nAs mentioned in the Introduction, humans have the decisive judgment about the received service quality. Therefore, scheduling strategies that exploit direct and conscious inputs of the users have the advantage of knowing their preferences and if they are satisfied with the service. On the other hand, these approaches require that end-user devices are capable of receiving the necessary user inputs. The strategies described next (and summarized in Table 4) adopt this active user approach. Aristomenopoulos et al. (2010) presented a QoE provisioning method that enables users to indicate their preference in a dynamic and asynchronous manner concerning the instantaneous perception of the service performance. To attain this goal, a Graphical User Interface (GUI) displays and captures the users' options (increase or reduce quality), the feasibility and the repercussions of their act (cost). When the preference of a user is manifested, the features of the utility functions are dynamically adjusted, in order to exploit the NUM theory by enabling the smooth incorporation of users' subjective decision in the resource allocation process. A QoE-aware scheduling framework is proposed in (Lee et al., 2014) to maximize the average amount of satisfied users, where it is assumed that users can feedback one bit to express their degree of satisfaction. Since the decisions taken by the users have a direct influence on the QoE enhancement process, non-trivial fairness constraints are also added so as to prevent starvation.\n\n## 4.4. Other QoE-oriented Scheduling Methods\n\nAll the previous scheduling algorithms addressed the downlink scenario, with the goal of maximizing the aggregate experienced quality of all users within one cell of a single operator. Nonetheless, QoE-aware scheduling methods can enhance the wireless resources management in other scenarios, such as the uplink direction, the multicell case, under heterogeneous, cognitive radio, relay and MIMO networks, as well as when dealing with energyrelated issues.\n\n## 4.4.1. Uplink\n\nA resource scheduling framework for live video uplinking is proposed in (Essaili et al., 2011), in which more resources are allocated for popular contents while ensuring a certain QoE level (based on throughput) for the less popular ones. Wu et al. (2012) presented a QoE-driven scheduler for uplink unicast video delivery (intended for surveillance systems), in which the required throughput and delay are the main factors taken into account by the resource allocation scheme. In (Song et al., 2014), a QoE-based joint resource allocation method is proposed for uplink scenario of LTE networks that make use of carrier aggregation, where the assessment of the users' QoE satisfaction degree is performed by a resource cost, link reward and utility function designed by the authors. The works presented in (Condoluci et al., 2017;Liu et al., 2018) address haptic teleoperation over wireless networks and introduce new allocations algorithms regarding the scheduling process in the uplink direction, which aim at improving the QoE by reducing the communication delay. With respect to disaster scenarios, and assuming that base stations are mounted on Unmanned Aerial Vehicles (UAVs), an uplink scheduling procedure is proposed in (Ranjan et al., 2018) that has the goal of enhancing the QoE of critical users (i.e., those that are in danger positions or have low battery power), so that they can better communicate with the outside world.\n\n## 4.4.2. Multi-cell\n\nThe scheduling strategies mentioned up to this point consider the situation where each base station serves as an independent scheduler. However, a centralized network controller can be adopted in order to improve the QoE of those users that can connect to more than one base station. The works presented in (Zheng et al., 2014;Cho et al., 2015) addressed the inter-cell interference problem and devised QoE-oriented resource allocation techniques that aim at enhancing the users' satisfaction and fairness, especially for cell edge users. Kim et al. (2015) proposed a QoE-aware scheduling algorithm that also incorporates admission and handover procedures (to neighboring base stations), in order to try to ensure that video streaming users that connected previously to the network maintain at least an acceptable QoE (namely by taking into account the buffer level). In (Miller et al., 2015), the resource allocation problem is considered for a high number of simultaneous video streaming sessions within a dense wireless network scenario, in which a central scheduler has the goal of providing the allocation configuration (among all base stations) that enables to improve the users' QoE.\n\n## 4.4.3. Heterogeneous, Cognitive Radio, Relay & Multi-\n\nuser MIMO Networks Only single operators have been considered so far, but some research has already been done regarding heterogeneous wireless networks, where users have the desire of being connected to the best accessible networks according to each user application specifications and personal preferences. For instance, Toseef et al. (2011) adopted a game-theoretic approach as a framework for user satisfaction based wireless resource scheduling with various service providers, different sorts of users and several service categories. In (Jailton et al., 2013), a QoE-based handover architecture is presented for heterogeneous mobile networks, which has the goal of providing seamless mobility in multi-operator and multi-access systems while ensuring that users have the best possible connection in terms QoE. Regarding mobile traffic offloading, a method to improve the QoE of video delivery, which incorporates a collaboration between Wi-Fi hotspots and LTE base stations, is proposed in (Seyedebrahimi and Peng, 2015), where the subjective quality assessment is based on video playback discontinuities. Morel and Randriamasy (2017) devised a QoE-based scheduling algorithm that aims at enhancing the video delivery over heterogeneous mobile networks, namely by limiting the inter-cell interference experienced by some users. The work presented in (Abbas et al., 2017) addresses the real-time traffic splitting across cellular and Wi-Fi heterogeneous networks (focusing on video streaming applications) and provides a solution for resource allocation that enhances QoE while reducing delay and energy consumption (namely of mobile terminals). Regardless of each specific scheduling solution for heterogeneous networks, all of them have a major difference when compared to the traditional scheduling methods: as depicted in Fig. 7, an extra entity is required -indicated in the figure as \"Coordinator\" -as well as extra signaling is needed, in order to ensure a proper QoE-oriented scheduling in all instances, namely when performing inter-network handover or traffic splitting. Since different networks can make use of different technologies, or can even be associated with different service providers, it is also not possible, within heterogeneous networks, to make use of a single scheduler and apply the traditional scheduling solutions. Hence, not only each network has its own scheduler, but also the \"Coordinator\" has the complex task of coordinating these schedulers (which usually includes conveying relevant QoE information among them), in order to enhance the users' QoE or maintain at least an acceptable QoE.\n\nDevice-to-Device (D2D) systems can also benefit from QoE-aware approaches. In a somewhat similar fashion to the heterogeneous case, D2D networks also require a \"Coordinator\" unit (as well as extra signaling) for an effective QoE-oriented scheduling. More specifically, not only this unit is responsible for collecting information sent wirelessly from the devices (thus the extra signaling is less re- liable than the wired one of heterogeneous networks), but it also performs important scheduling tasks, such as which and when D2D pairs can access the channel or which data should be conveyed by each device. In (Zhu et al., 2015a), a QoE-driven allocation technique is proposed for video streaming through D2D transmissions, where the wireless resource scheduling goal is to enhance the time-averaged quality of video streams transmitted over D2D communications, while constraining the number of stall events for each stream. Hong et al. (2017) devised a QoE-aware D2Dbased mobile task outsourcing, in which the cellular mobile equipments cooperate so as to perform computational intensive tasks; a special node within the cellular network is responsible for the D2D task scheduling, where the maximum wait time of each task is adopted as the main QoE influence factor. A management procedure is presented in (Biswash and Jayakody, 2018) for mode switching between base station approach and D2D communications, where QoE parameters are the basis for the switching decisions, as well as for the inclusion, or not, of a third-party terminal to assist a D2D transmission. In order to provide greater flexibility within D2D systems, Sawyer and Smith (2019) proposed the categorization of D2D users into different groups (based on their practical application or service), which enables to define a suitable utility function for each group, followed by a solution for QoE-oriented resource allocation, which is formulated as a dynamic Stackelberg game that considers multi-criteria decision making.\n\nWith respect to vehicular networks, some works provide QoE-driven frameworks, namely for video-on-demand over urban vehicular networks (Xu et al., 2013), for scalable video streaming over cooperative Vehicle-to-Vehicle (V2V) and Vehicle-to-Infrastructure (V2I) communications (Yaacoub et al., 2015), as well as with the goal of reducing the transmission delay of vehicular security applications (Ding et al., 2018).\n\nQoE provisioning schemes can also be used when addressing other wireless resources scheduling problems, such as in cognitive radio networks. It is noteworthy to men-  tion that, on the one hand, the opportunistic spectrum access of cognitive radio approaches can lead to an increase of the total available wireless resources that can be scheduled for the secondary users, but, on the other hand, the resources allocated to the secondary users can be reoccupied by the primary users at any time. Hence, not only this leads to an unstable availability of wireless resources for the secondary users (as depicted in Fig. 8), but also this dynamic behavior poses an additional challenge to the wireless system scheduler. Jiang et al. (2012) developed a channel assigning algorithm for the transmission of multimedia content over cognitive radio systems, where the base station allocates available channels to secondary users according to the respective QoE requirements (delay and multimedia content quality). A jointly design of spectrum sensing and access policies for multi-user QoEoriented video delivery within cognitive radio networks is presented in (He et al., 2016), in which the goal is to achieve fairness among the users while maximizing the average QoE (which is based on throughput). Zhang et al. (2017) addressed networks where different types of base stations are deployed to exploit a heterogeneous spectrum pool, containing licensed and harvested spectrum, in order to propose a game-theoretic approach that solves the problem of optimizing the global users' satisfaction (based on their throughput requirement) by jointly optimizing spectrum sharing, user scheduling, and power allocation in a decentralized manner. A framework is devised in (Piran et al., 2017) that has the goal of managing the inevitable spectrum handoff within cognitive networks while providing seamless multimedia content streaming and QoE enhancement (namely by taking into consideration the delay and the quality of the multimedia content). In (Lin et al., 2017), a QoE-oriented dynamic channel access procedure is presented in order to handle the sharing of spectrum between licensed and secondary users; the scheduler takes into account the type of service that is being requested by the secondary users (either delay sensitive or not) and aims at minimizing the average queuing time of the re-spective packets. Yin et al. (2019) proposed a QoE-driven rate control and resource allocation scheme for cognitive Machine-to-Machine (M2M) communication; the authors focused on how to maximize the QoE of all M2M pairs, namely by employing a stochastic optimization model in which the user-perceived application quality metric is associated with data rates.\n\nWith respect to wireless relaying in cognitive radio networks, and by considering QoE indicators (relay buffer status) rather than traditional QoS indicators, Wu et al. (2013) showed that better user experience can be achieved with sub-optimum system capacity. Reis et al. (2010) considered multi-hop wireless networks and introduced a scheduling algorithm that jointly performs an optimization of multiple video, audio and data streams transmission based on certain QoE targets. The works presented in (Xiang et al., 2017;Fan and Zhao, 2018) developed crosslayer resource allocation schemes for two-hop and ad hoc networks, respectively, where the goal of both works is to enhance the QoE by minimizing the end-to-end video delivery time. In (Bethanabhotla et al., 2016), an efficient system is proposed for video streaming over a wireless system composed by a large amount of wireless helper equipments with multi-user MIMO capabilities, where QoE metrics like video quality and rebuffering percentage are used to optimize the transmission scheduling of users at each base station. Other examples of QoE-based scheduling for multi-user MIMO systems are given in (Cao et al., 2012;Chen et al., 2017), where user selection procedures based on transmitted rate and delay are proposed in order to maximize the average multi-service satisfaction degree. With respect to Virtual Reality (VR) over multiuser MIMO networks, Huang and Zhang (2018) devised a resource allocation technique based on a maximum aggregate delay-capacity utility function, which aims at reducing the delay between user motion and the presentation of multimedia content, as well it aims at maximizing the number of connected VR users with an acceptable QoE.\n\n## 4.4.4. Energy-related Issues\n\nThe \"green\" networks is also a domain where QoEbased wireless resources management can have an important role. For instance, Ma et al. (2012) proposed a framework to optimize the power allocation at the base station side, in which the adopted utility function -formulated as\n\nwhere p i,n stands for the allocated power to the i th user on the n th sub-channel and M OS i denotes the MOS of the i th user -aims at providing a balance between energy efficiency and QoE for video streaming users. Li et al. (2012) devised a power allocation optimization method regarding the QoE for multi-services, which is intended to solve the following problem (where the goal is to minimize the overall power consumption, while ensuring a good level of perceived quality to each user):\n\nwhere P total represents the total power allocated to the users, P max corresponds to the system maximum power, Ψ i (•) denotes the mapping relationship between p i,n and a MOS value regarding the multimedia service chosen by the i th user, and M OS i min represents the minimum acceptable MOS also concerning the multimedia service chosen by the i th user. In (Gabale and Subramanian, 2014), an \"energy-source\" based method for delivering multimedia content asynchronously is presented, which adjusts the consignment of delay-tolerant content according to the time intervals where the renewable energy is available, i.e., it tries to balance the service provider energy costs and the QoE (namely the delay) experienced by users. An approach to power-cycle base stations and to control the playback of video streams is presented in (Draxler et al., 2014), where the goal is to reduce the overall energy consumed by the base stations while maintaining a high QoE for the users. A framework to reduce small cell energy consumption contingent on QoE restrictions has been proposed in (Sapountzis et al., 2014), where an analytical investigation is performed regarding the trade-off between switching off underloaded base stations and the performance degradation experienced by the users. An energy efficient resource on-off switching framework for a cellular network comprising a femtocell at the cell edge of a macrocell is investigated in (Farrokhi and Ercetin, 2016), where the goal is to minimize the energy consumption of the cellular network while satisfying a desired level of QoE, which is defined as buffer starvation probability of a mobile device. In Kotagi and Murthy (2019), an energy-efficient scheduling technique is devised in order to decrease energy consumption of wireless systems, such as heterogeneous networks and dense femtocell networks, as well as to mitigate the cell edge interference, in which a QoEdriven strategy considers resources demand at every pool (instead of solely depending on the signal-to-noise ratio) before shifting a user from one pool to another. Xu et al. (2019) proposed a resource and re-association scheduling method based on Benders' decomposition to decrease the energy consumption within Wireless Local Area Networks (WLANs), namely by aggregating users to fewer Access Points (APs) and by turning off many APs without compromising the users' QoE as well as the network coverage. With respect to terminals energy consumption, in (Csernai and Gulyas, 2011), a framework is presented that decreases the energy used by mobile equipments during video delivery over WLANs, in which a QoE-based algorithm makes an estimation of the video quality perceived by a user and adapts the sleep periods of the wireless device so as to enhance the power efficiency while preserving video quality. Ksentini and Hadjadj-Aoul (2012) proposed a novel approach for energy conservation in mobile terminals regarding VoIP flows over WLANs, namely by tuning the sleep intervals according to the user QoE. Schedulers that make use of the discontinuous reception method for LTE systems have been proposed in (Szabó et al., 2014;Mushtaq et al., 2015) (with the latter focusing on VoIP applications), which can provide battery saving for the terminals without degrading the QoE of the services, namely the delay perceived by a user. In (Hong and Kim, 2019), a scheduling policy is proposed to save energy of mobile devices via optimal transmission scheduling of mobile-to-cloud task offloading, which considers several QoE domains in order to capture the energy-latencypricing trade-off. Gao et al. (2019) also addressed computation offloading but with respect to UAV cloud system, namely by devising a resource allocation algorithm with heterogeneous QoE support which is able to enhance the energy efficiency of the UAVs. As previously mentioned, Abbas et al. (2017) proposed a method that reduces energy consumption of mobile terminals regarding cellular and Wi-Fi heterogeneous networks.\n\n$$U i n p i,n = M OS i n p i,n,(33)$$\n\n$$minimize P total = i n p i,n subject to Ψ i n p i,n ≥ M OS i min P total ≤ P max p i,n ≥ 0, ∀ i,n,(34)$$\n\n## 4.4.5. General Considerations on Other QoE-oriented\n\nScheduling Methods The scheduling strategies mentioned in this subsection, which are also summarized in Table 5, clearly show that QoE-aware decisions are useful in a vast field of resource management regarding wireless communications. On the one hand, these scenarios (other than the downlink direction with respect to a single base station) introduce some particularities that must be taken into account if one wants to adopt the techniques that were presented in Sections 4.1, 4.2 and 4.3 -issues like, e.g., the exchange of information between different wireless technologies when heterogeneous networks are considered, or the dynamic nature of cognitive radio networks, among others, can lead to a more challenging scheduling scenario. On the other hand, the same particular characteristics can also provide new tools that enable to leverage the users' QoE -for instance, the handover procedure that is available for the centralized approach of the multi-cell scenario (namely for those users that can be served by more than one base station simultaneously) can be activated with the purpose of liberating resources from a base station, thus enabling that additional users are able to achieve a higher QoE.\n\nStill within the scope of other QoE-oriented scheduling methods, it is noteworthy to mention that the QoE might also be improved if some scheduling decisions are conveyed to the users, such as by giving the option of saving energy at the terminal side at the cost of, e.g., a slightly inferior video streaming quality. The scheduling methods referred above perform resource allocation and/or prioritization at the wireless MAC/physical layer level, which is the scope of this work. Nevertheless, there are also QoE-aware algorithms that, although taking into consideration the wireless channel specificities, they permit to preserve an entirely standard and application-layer unaware physical/MAC operationfor instance, algorithms that run above the MAC layer and which arrange suitably the order of the data units delivered to this layer; more examples and details can be found in (Bianchi et al., 2010;Ramamurthi et al., 2014;Chen et al., 2015b;Radics et al., 2015;Borkowski et al., 2016;Eswara et al., 2016;Héder et al., 2016;Kumar et al., 2016;Tajima and Okabe, 2016;Triki et al., 2016;Zheng et al., 2017;Zhang et al., 2019;Kim and Chung, 2019).\n\nTaking into account the scheduling strategies surveyed herein, video streaming is clearly the application that can benefit more from QoE-aware schedulers. In addition, this will keep being an important topic of research, since not only video streaming is now the most popular application on the Internet, having been responsible for 59% of the entire mobile data traffic at the end of 2017, but also it is foreseen that over three-fourths (79%) of the worldwide mobile data traffic will be video by 2022 (Cisco, 2018). Moreover, as depicted in Fig. 9, global mobile data traffic will grow 7-fold from 2017 to 2022, a compound annual growth rate of 46%, thus reaching 77.5 exabytes per  month by 2022, up from 11.5 exabytes per month in 2017, which means that mobile data traffic will grow 2.0 times faster than fixed traffic from 2017 to 2022. On top of that, some authors already suggested that multimedia services should be priced based on the QoE rather than based on the transmitted binary data (Wang and Wang, 2018).\n\nNevertheless, it was also seen that a scheduling algorithm and its underlying QoE estimation model need to be careful adjusted for each application, since the user's perception regarding different applications is influenced by different factors. In other words, a \"one-fits-all\" solution for wireless resource schedulers is not feasible when QoE comes into action, as quality expectations are highly service dependent, which means that scheduling strategies should always take into account the intended application. As a rule of thumb, schedulers should try to avoid buffer emptiness when dealing with video streaming, they should reduce the delay as much as possible for VoIP and web browsing services, whereas if they are unaware of the application, then serving the best possible throughput for each user is the most reasonable option.\n\nAnother issue that follows the previous considerations is that the performance of different QoE-oriented schedulers might not be easily comparable, mainly due to the fact that a common reference scenario might be impractical when dealing with different applications. Accordingly, it is important to clearly identify which are the specific goals of each solution, in order to determine which ones can be adopted for the same scenario; otherwise, the performance comparison becomes unfair, even for applications that appear to be similar -for instance, and considering the video streaming application, a QoE-based scheduling algorithm that was expressly adjusted to handle the transmission of certain videos, namely those that are stored in a repository, might underperform in other video streaming scenarios, e.g., live transmissions.\n\nIt is noteworthy to point out that admission control was not taken into account by almost all the scheduling algorithms surveyed herein (the work of Kim et al. (2015) is the only exception). Admission control can play an important role in terms of QoE provisioning, namely by admitting a new user only if the already connected users and the new user can have an acceptable QoE. Although it seems advantageous to jointly design QoE-oriented scheduling strategies and admission control mechanisms, the vast majority of the scheduling methods referred above only studied the impact of the number of users on the QoE; nevertheless, these studies could be used to infer the number of users that can be accommodated in the wireless network, thus serving as an input when devising admission control procedures. On the other hand, only a few authors have presented admission control solutions that are QoE-aware -cf. (Chen et al., 2015;Zhou et al., 2015;Qadir et al., 2015;Ammar and Varela, 2015;Ksentini et al., 2016). One possible explanation for the lack of literature about this topic is the fact that, as found by Ammar and Varela (2016) (which evaluated and compared QoE-based admission control mechanisms), there is an inherent difficulty to precisely calibrate the algorithms in order to obtain satisfying QoE-oriented admission decisions, namely because not only the best calibration, but also the overall performance of the admission control algorithms varies strongly with the underlying scenario.\n\nWith respect to the three categories of QoE-aware schedulers addressed in this work, there is still room for further developments in all of them, but the active enduser device / passive user strategies are the ones which might be the focus of attention in the future. The main reason for this is that they enable to gather the maximum amount of relevant QoE information as close as possible to the end-user without requiring direct and conscious human QoE inputs; accordingly, they are the clear candidates to better enhance the performance of QoE-oriented schedulers without annoying the end-user with QoE assessment prompts. Moreover, with the support of miscellaneous biometric sensors that are being incorporated in smartphones -e.g., the TrueDepth camera system of iPhone X (Huggins, 2017), which is able to analyze more than 50 different facial muscle movements, including attention confirmation by detecting the direction of the user's gaze -more accurate QoE data are becoming available at the end-user device that stem from live human biometric indirect inputs -like pupil variations and galvanic skin reactions (Shye et al., 2008), facial expressions (Whitehill et al., 2008), and body gesticulations (Castellano et al., 2007) -which can provide a viable online QoE optimization scheme in a model-free manner (Du et al., 2017). Nonetheless, and in spite of the fact that there are already several feedback mechanisms that allow to report a significant number of parameters from the end-user devices to the scheduler, as in the 3GP-DASH standard, important challenges still arise: which are the relevant QoE parameters and which is the specific relationship between each one of them and the user's subjective perception? Within this subject, machine learning methods can play a major role, higher than the one seen so far, as they enable to devise complex models and learn about non-obvious correlations between the collected parameters and the users' QoE. In addition, these computer science techniques are a powerful tool to better deal with particular issues regarding mobile networks, such as phone battery, phone overheat, or data connectivity costs, which make QoE assessment more demanding when compared to fixed networks.\n\nLast but not least, QoE evolves over time in the same manner as technology itself does: for instance, and as illustrated in Fig. 10, although today a user may require super high definition video content in order to say that he is experiencing a great QoE, some decades ago the same subjective perceived quality would be true for a TV transmission with standard definition. Consequently, a scheduler that takes into account a QoE model that reflects today's reality may not be the most appropriate one in the future, meaning that QoE-oriented schedulers will always attract the research community's attention.\n\n## 6. Conclusions\n\nWireless resources scheduling is starting to become more user-centric QoE-oriented, replacing the traditional system-centric QoS-driven approach. Thus, the network operators, in addition to multimedia service providers, aspire to have faithful models that are able to evaluate, foresee and even manage QoE, specially for multimedia content transmissions. This paper provided an extensive survey about this research topic: first, QoE was explained,  namely the factors that influence the user experience, as well as some QoE estimation methods regarding multimedia services over communication systems; next, the evolution of wireless scheduling techniques was presented, including QoS-QoE mapping strategies and utility-based optimization; finally, state-of-the-art QoE-based scheduling strategies for wireless systems were described, highlighting the application/service of each solution, as well as the parameters adopted for QoE optimization. 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+    },
+    "test": {
+      "total_tokens": 54990599,
+      "example": "# Mitigating Unnecessary Handovers in Ultra-Dense Networks through Machine Learning-based Mobility Prediction\n\nDonglin Wang, Anjie Qiu, Sanket Partani, Qiuheng Zhou, Hans Schotten\n\n## Abstract\n\nIn 5G wireless communication, Intelligent Transportation Systems (ITS) and automobile applications, such as autonomous driving, are widely examined. These applications have strict requirements and often require high Quality of Service (QoS). In an urban setting, Ultra-Dense Networks (UDNs) have the potential to not only provide optimal QoS but also increase system capacity and frequency reuse. However, the current architecture of 5G UDN of dense Small Cell Nodes (SCNs) deployment prompts increased delay, handover times, and handover failures. In this paper, we propose a Machine Learning (ML) supported Mobility Prediction (MP) strategy to predict future Vehicle User Equipment (VUE) mobility and handover locations. The primary aim of the proposed methodology is to minimize Unnecessary Handover (UHO) while ensuring VUEs take full advantage of the deployed UDN. We evaluate and validate our approach on a downlink system-level simulator. We predict mobility using Support Vector Machine (SVM), Decision Tree Classifier (DTC), and Random Forest Classifier (RFC). The simulation results show an average reduction of 30% in handover times by utilizing ML-based MP, with RFC showing the most reduction up to 70% in some cases.\n\n## I. INTRODUCTION\n\nThe fifth generation (5G) of cellular communication technology aims to achieve a significant enhancement in communication capacity [1]. One of the key innovations in 5G is the implementation of UDN, which involves the deployment of a high density of small cells in order to support the efficient utilization of the spectrum and augment the coverage and capacity of cellular networks. Therefore, wide attention has been paid to the application of UDNs. VUEs have a collection of flexible mobile access options provided by UDN with higher spectrum efficiency through the extensive deployment of SCNs [2].\n\nThe UDN network consists of a large number of small cells that can support spatial reuse of the spectrum to increase the coverage and capacity of cellular networks remarkably [2]. As shown in Fig. 1, a simplified UDN network has multiple SCNs, a macro Base Station (BS), a network server, a network, and a moving VUE. Densely deployed SCNs improve the frequency reuse and system capacity, and more VUEs and mobile users Fig. 1. A showcase of a UDN network can be supported by UDN [3]. Nonetheless, the deployment of small cells differs from that of traditional BSs, and the mobility in UDN poses unique challenges in comparison to previous multi-tier networks. UDNs are faced with a plethora of obstacles, including increased interference, more frequent handovers, delays in traffic delivery, radio link failures, and elevated costs [4]. Literature has been devoted to providing insights on the challenges associated with UDN deployment and identifying potential solutions [5]  [6]. In this work [7], the UDN security problem is considered. The authors proposed an Implicit Certificate (IC) scheme that is expected to solve the UDN security problem among the Access Point (AP) in a dynamic APs group and between the AP and User Equipment (UE).\n\nIn [8], the Third Generation Partner Program (3GPP) has defined the handover processes protocol. Handover events are generally triggered based on periodic measurement reports of VUEs and handover parameters, i.e., Time-To-Trigger (TTT) and Handover Hysteresis value (Hys) [3]. In our previous work [3], the authors analyze the effect of TTT, density, and velocity on the performance of handovers. In [9], the authors proposed a handover optimization method that uses fuzzy logic to reduce handover probability. The simulation results show handover probability and handover Ping-Pong (PP) effects have been significantly reduced. In [10], Shi et al. proposed a user MP method based on the Lagrange Interpolation. By using the slope of the trajectory polynomial and velocity, the transition probability of VUEs is detected. However, this method suffers from low mobility prediction accuracy and less UHO reduction. In [11], the authors tried to minimize the handover failures and PP rate by introducing an intelligent dynamic handover parameter optimization approach. With the development, more and more ML algorithms are used in 5G communication [12]. We propose ML algorithm-based MP to reduce the UHO times. Different algorithms are used and compared the prediction results. Then the predicted results are applied to reduce handover times.\n\nThe remaining sections of this paper are organized as follows: Section II delves into the implementation of MLbased MP and provides an overview of the system model, the ML models employed, and the results obtained. Section III discusses the handover process in UDNs with and without MP in depth. Section IV introduces the system simulator and presents a comprehensive analysis of the results obtained. Finally, in Section V, the study concludes with a summary of the findings and suggestions for future research.\n\n## II. MACHINE LEARNING-BASED MOBILITY PREDICTION\n\nWith the advancement of modern society, mobile communication is increasingly required to operate in highly dynamic, pervasive computing environments on the road network. This demands increased capacity and exceptional QoS. To meet these requirements, ML-based MP has emerged as a critical enabler in mobile communication. MP utilizes historical traffic information to predict the future locations of VUEs. thereby enabling efficient radio resource management, route planning, vehicle dispatching, and reducing traffic congestion and handovers [13]. In the following subsections, a detailed examination of ML-based MP will be provided, including the system model and various ML algorithms employed.\n\n## A. System models\n\nAs depicted in Fig. 2, a simulation scenario of UDN with SCNs is constructed. The deployment of SCNs in UDN adheres to the Poisson Point Process (PPP) distribution model [14]. It's assumed that all SCNs possess identical characteristics, including antenna height, antenna tilt, and others.\n\nIn this system model, three distinct routes are available for thousands of VUEs within a 1000-meter *1000-meter area [15]. Fig. 3 illustrates the distribution of VUEs on potential routes, varying mobility demands depending on the time of the day. The simulation scenario and moving VUEs are generated using SUMO simulation software [16]. VUEs are classified after passing junction A, i.e., the VUEs on Route 1 are marked with the ID \"0\". VUEs on Route 2 are identified with the ID \"1\", and the VUEs on Route 3 is labeled with the ID \"2\".\n\n## B. Dataset for training\n\nEach vehicle is simulated for 70000 milliseconds, and the contextual information of each vehicle, such as time step, route ID, longitude and latitude of VUEs, is collected. The generated dataset D = (x i, y i, r i, t i ) is utilized to train various ML algorithms, where x i represents longitude, y i represents latitude, r i represents the route ID, and t i represents the time step. The dataset D is randomly partitioned into a training dataset (75%) and a testing dataset (25%). The size of our dataset is N = 82435 obtained from the SUMO simulation software.\n\n## C. Machine learning algorithms\n\nThe process of MP is illustrated in Fig. 4. In this process, the collected dataset D is used to identify the relationship between input data and the desired output value. When new input data (x k, y k, t k ) is input into the model, the model can predict the next output value of route (r k ) that V U E k will take at the time slot t k. As the MP problem is a classification problem, a variety of ML models such as SVM, DTC, and RFC are employed for MP and compared in terms of performance for large-scale data training.\n\n1) Support Vector Machine (SVM): SVMs are a widely used ML algorithm in classification and regression. As stated in [17], they have been widely adopted in the field of MP of VUE due to their ability to achieve high prediction accuracy while minimizing the risk of overfitting the data. In this study, we utilized the SVM algorithm to predict the VUE's route as previously demonstrated in [13].\n\n2) Decision Tree Classifier (DTC): DTCs are a popular ML algorithm that aims to simplify the complex decisionmaking process by breaking them down into a series of simpler decisions, hoping the final solution obtained this way would resemble the intended desired solution [18].\n\n3) Random Forest Classifier (RFC): RFCs are an ensemble technique that combines multiple decision trees to improve prediction accuracy. According to [19], the technique works by having each decision tree vote for the most popular class, and the class that receives the majority of votes is chosen as the final output. This approach has been shown to result in significant improvements in classification accuracy when compared to using a single decision tree.\n\n## D. Performance comparison of ML algorithms\n\nIn Table I, the performance comparison of three ML models is illustrated, in terms of five performance metrics: Training Set Score (TSS), Testing Set Score (TESS), Balanced Accuracy Score (BAS), Recall Score (RS), and Precision Score (PS). The TSS and TESS values indicate the model's performance on the training and testing sets, respectively. A comparable TSS and TESS suggest that there is no significant overfitting or underfitting. The BAS metric is used to quantify the model's overall performance by taking into account the recall obtained on each class, while the RS and PS metrics measure the model's ability to identify all positive samples, and avoid falsely labeling negative samples as positive respectively. SVM, DTC, and RFC used in this system all have a very high TSS with 0.9112, 1.0, and 1.0 respectively, indicating that the models have been trained very well on the data, and TESS values are also high, indicating that the models are able to generalize well to new data. Especially, RFC with the highest TESS value of 0.9606.\n\nThe trained ML models are used to predict the user route based on the historical data, for various time periods. Table II shows the TESS values (denoted as prediction accuracy) for the three different ML models. As an example, during the morning time period of 7-8 AM, the probability of a VUE taking Route 1 is 0.9113 when using the SVM model as opposed to 0.9063 when using the RFC model. The prediction results are used in the following sections for reducing handover. The integration of SCNs in UDN has led to a proliferation of UHO. To mitigate this issue, we propose a new method that utilizes MP to reduce handover times. For comparison, the conventional 5G handover method is introduced as well. Simulation results of two methods are used to compare the performance on the handover.\n\n## A. Conventional 5G handover calculation\n\nThe conventional 5G handover calculation method is well-established, in which a user explores the Signal-to-Interference-plus-Noise Ratio (SINR) and availability of all nearby SCNs to discover or search for potential SCNs to connect to [20] [21], as depicted in Fig. 2. In this figure, the blue circle represents the communication range (R) between VUE and SCNs. If the distance (D ki ) between the V U E k and the i th SCN is less than or equal to the communication range, then the i th SCN will be considered as a candidate SCN. In total, T candidates are available including the SCNs marked in orange and blue in Fig. 2. The SCN that provides the highest SINR value will become the target SCN, and the VUE will be scheduled to connect to it, even though it may not be necessary. However, the highly dense deployment of SCNs in UDN leads to more UHO, thus increasing traffic delay, interference, radio link failure, and ultimately decreasing the user experience quality.\n\n## B. 5G handover calculation with Prediction\n\nThe proposed method for 5G handover optimization incorporates the utilization of predictive techniques to anticipate handover events. The method involves the implementation of a screening distance and screening angle, calculated as D = 300 meters and θ = 50/ √ V + 18 respectively, where V represents the velocity of the VUE [10]. The SCNs within the defined screening distance and angle are considered as predicted candidates for handover (denoted as N ) in orange, with the remainder of the SCNs within the communication range referred to as unpredicted candidates (denoted as M ).\n\nAs illustrated in Fig. 2, if a V U E k has the highest SINR value (denoted as ki dBm) from an unpredicted candidate (M i ) at a future position, the unpredicted candidate is labeled as the incorrect SCN. The highest SINR (denoted as kj dBm) value from the predicted candidate (N j ) is then compared to the SINR value (denoted as kx dBm) of the V U E k from the current serving SCN (x). An SINR offset value (denoted as so dBm), calculated as so dBm = (ki dBmmax(kj dBm, kx dBm)) * 3 is employed. Then the highest SINR (ki dBm) from the incorrect SCN becomes ki dBm = ki dBmso dBm to ensure that the V U E k doesn't switch unpredicted candidate (M i ) but switches to the predicted SCN (N j ) with the highest SINR kj dBm, thus optimizing the user experience and reducing handover times.\n\nThe proposed method for 5G handover triggering as 3GPP standardization is outlined in Algorithm I.\n\n## Algorithm 1 Handover triggering logical algorithm\n\nInput: serving scn, serving sinr, best sinr, target scn, sinr min, avg sinr, best cio, current cio, ho hys, ttt, ho timer, ho trigger, ho exec time Output: ho times  III. As demonstrated in Algorithm I, the total number of successful handovers in each simulation is extracted. The handover triggering process is executed in the following steps: 1) It is essential to ensure that the target scn is distinct from the current serving scn, meaning that they possess different identifiers and are located in disparate locations. This step ensures that the handover process is triggered only when a distinct target cell is identified, rather than the current serving cell, thus avoiding UHOs and ensuring optimal network performance; 2) This step involves evaluating the value of best sinr, which represents the best SINR value calculated previously, against a pre-defined threshold of sinr min = -7 dB. Additionally, the difference between best sinr and avg sinr pluses the difference of best cio and current cio must be greater than the handover hysteresis value of ho hys = 3 dB. The values of best cio and current cio are initially set to 0, however, they may be affected when a load balancing algorithm is implemented. avg sinr represents the average SINR of the VUE in relation to the current SCN and is calculated using the previous X SINR values, where X represents the number of previous SINR values used, and taking their average. In this simulation, a counter is utilized to hold 10 previous SINR values and avg sinr is calculated at each tic in the simulator. It can only be calculated when the previous X SINR values are available; 3) and 4) If all the conditions mentioned in the previous steps are satisfied, this step sets the ho trigger flag to one, indicating that a handover should be triggered. Additionally, the ho timer counter is incremented by one; 5) This step involves evaluating whether the value of the ho timer counter has reached the predefined TTT value. If the TTT value has been met, it indicates that the handover should be executed. If the TTT value has not been met, the handover process will be delayed until the TTT value is reached. The TTT value is a crucial parameter that affects the handover rate, as it determines the time interval between the initiation of the handover process and its execution; 6) This step marks the execution of the handover process. The serving scn is updated to equal the target scn, indicating that the handover has taken place. The ho exec time is set to 25 tics, which represents the duration of the handover process. The output counter ho times is incremented by 1 to keep track of the number of successful handovers. The ho trigger and ho timer are reset to 0 to prepare for the next handover process. In case one of the conditions mentioned in the previous steps is not satisfied, the algorithm will be reexecuted. The above description outlines the basic logic for the handover triggering process in each simulation.\n\n## IV. SIMULATION RESULTS\n\nA system-level simulator environment is developed using Python to set up the simulation scenario. The simulation parameters can be found in Table III. The simulation is run for 100 iterations, with the total number of successful handover times being recorded. The density of the SCNs is set at 50 scns/km. The simulation results are used to evaluate the effectiveness of the proposed handover optimization method in reducing the number of handovers and improving the overall network performance.\n\nThe results of the simulation are illustrated in Fig. 5, which presents the handover times for different TTT values when using three different ML algorithms (SVM, DTC, and RFC) and a fixed VUE velocity of 10 km/h. Additionally, the handover times for the case without the use of any ML algorithms are provided for comparison.\n\nStarting with a TTT value of 1, the handover times for the cases without MP, with MP using SVM, and with MP using DTC are 200. However, only with MP using RFC, the handover times are reduced to 100 times. As the TTT value increases to 2 tics, the differences in the application of MP with various ML models become more apparent. It can be observed that when the TTT value is relatively small, MP based on SVM and MP based on DTC do not significantly affect the handover times in reducing them. The simulated results in Fig. 5 demonstrates that the TTT value plays an important role in the handover reduction. Furthermore, it is evident from the results that as the TTT value increases, the number of handovers reduces significantly, even approaching 0 at 12 tics which means VUE fails in radio connection. This trend can be observed regardless of the application of any ML model, as the VUEs experience a substantial degradation of SINR during the TTT period. This implies that there is an optimal TTT value for a given VUE velocity that can be used to reduce UHO while avoiding handover failure. To compare the effect of MP using ML algorithms and without using ML algorithms on handover performance, when the TTT value is set to 4, the handover times of the traditional method without MP is 179 times. However, when the results of MP based on ML algorithms are considered, the handover times have decreased significantly. For example, when the MP result based on SVM is applied to the simulation, the handover times are reduced to 103, which is almost the same as the handover times obtained using the result of DTC-based MP (112). In Fig. 6, the effect of varying VUE velocity on handover times is presented. As the velocity increases, the handover procedure becomes more frequent and rapid, making the handover performance more time-critical, especially for real-time services. When no ML algorithms are used in the simulation, and only traditional handover is considered, the handover times increase from 200 times at 10 km/h to 700 times at 50 km/h, as indicated by the red bars. However, when the RFC algorithm is used, the handover times are reduced as compared to the conventional protocol, from 100 times at 10 km/h to 400 times at 50 km/h. This demonstrates that applying ML algorithms to MP can reduce UHO and save time cost. It is also worth noting that when the vehicle speed is 10 km/h, there is no difference between without ML and with MP using SVM and MP using DTC. This is because, at lower speeds, the handover process is less frequent than in the case of higher speeds, which also shows that MP has a less significant impact on the switching time when the speed of the VUE is relatively low. The reductions in UHO obtained by applying MP results based on three ML algorithms against traditional handover without any ML are shown as a line chart in Fig. 6. The average handover reduction ratio for SVM, DTC, and RFC are 29.44%, 34.93%, and 56.75% respectively. When using the RFC algorithm, the greatest decrease in handover times of 71.43% is obtained at a VUE velocity of 40 km/h. On an average, the use of ML algorithms in MP leads to an reduction of 30.28% in handover times. Overall, the results of the simulation demonstrate that the proposed handover optimization method using MP with different ML algorithms can significantly reduce the number of handovers and improve the overall network performance.\n\n## V. CONCLUSION\n\nIn summary, the proposed MP method based on ML algorithms has been shown to be effective in reducing UHO in a UDN scenario. By predicting the future route of a VUE, the system can anticipate potential handovers and make more informed decisions to reduce the number of handovers and improve the overall network performance.\n\nThree ML algorithms, SVM, DTC, and RFC, were evaluated and compared for their performance in the simulator. The results show that the RFC algorithm has the highest prediction accuracy and provides the most significant reduction in UHO. In addition, the simulation results also show that the handover performance is affected by various factors such as the TTT and VUE velocity. It is found that larger TTT values may lead to handover failure, but there is a suitable TTT for a given VUE velocity that can also help reduce UHO. Also, the handover times increase with rising velocities, making the handover performance more critical for high-speed VUEs.\n\nIn conclusion, the proposed MP method based on ML algorithms can effectively reduce UHO with an average decrease of 30.28% in handover times and improve the QoS for realtime applications in 5G cellular networks. It is recommended to implement different ML models in different use cases to achieve variant and comparable results. In future work, we plan to implement a more realistic simulation scenario with real data to see the reduction in UHO by MP-based ML algorithms and also to apply the ML algorithms on topics for 6G.\n\n## References\n\n1. Alsharif, Rosdiadee (2017) \"Evolution towards fifth generation (5g) wireless networks: Current trends and challenges in the deployment of millimetre wave, massive mimo, and small cells\" *Telecommunication Systems*\n\n2. López, Pérez, Ming et al. (2015) \"Towards 1 gbps/ue in cellular systems: Understanding ultra-dense small cell deployments\" *IEEE Communications Surveys & Tutorials*\n\n3. Wang, Qiu, Zhou et al. (2022) \"The effect of variable time to trigger, density, and velocity on handover of 5g nr ultra dense network\"\n\n4. Yu, Hansong, Hanlin et al. (2016) \"Ultradense networks: Survey of state of the art and future directions\"\n\n5. Gotsis, Stelios, Angeliki (2016) \"Ultradense networks: The new wireless frontier for enabling 5g access\" *IEEE Vehicular Technology Magazine*\n\n6. Hao, Xiao, Yu et al. (2016) \"Ultra dense network: Challenges enabling technologies and new trends\" *China Communications*\n\n7. Chen, Shanzhi, Hui et al. (2018) \"A security scheme of 5g ultradense network based on the implicit certificate\"\n\n8. Gpp \"Evolved Universal Terrestrial Radio Access\"\n\n9. \"3rd Generation Partnership Project (3GPP)\" *Radio Resource Control (RRC)*\n\n10. Alraih, Rosdiadee, Ibraheem et al. (2021) \"Ping-pong handover effect reduction in 5g and beyond networks\"\n\n11. Shi, Yunfeng, Liang (2019) \"A user mobility prediction method to reduce unnecessary handover for ultra dense network\"\n\n12. Huang, Mengting, Zongchang et al. (2022) \"Self-adapting handover parameters optimization for sdn-enabled udn\" *IEEE Transactions on Wireless Communications*\n\n13. Morocho, Manuel, Haeyoung et al. (2019) \"Machine learning for 5g/b5g mobile and wireless communications: Potential, limitations, and future directions\" *IEEE access*\n\n14. Wang, Zhou, Partani et al. (2021) \"Mobility prediction based on machine learning algorithms\"\n\n15. Last, Mathew (2017) \"Lectures on the poisson process\"\n\n16. Kuruvatti, Sachinkumar, Bavikatti et al. (2020) \"Mobility awareness in cellular networks to support service continuity in vehicular users\"\n\n17. Behrisch, Laura, Jakob et al. (2011) \"Sumo-simulation of urban mobility: an overview\"\n\n18. Jakkula (2006) \"Tutorial on support vector machine (svm)\"\n\n19. Safavian, David (1991) \"A survey of decision tree classifier methodology\" *IEEE transactions on systems, man, and cybernetics*\n\n20. Breiman (2001) \"Random forests\" *Machine learning*\n\n21. Tayyab, Xavier, Riku (2019) \"A survey on handover management: From lte to nr\" *IEEE Access*\n\n22. Peltonen, Ralf, David (2021) \"A comprehensive formal analysis of 5g handover\"<|endoftext|>"
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+      "example": "# Non-collective excitations in low-energy heavy-ion reactions: applicability of the random-matrix model\n\nS Yusa, K Hagino, N Rowley\n\n## Abstract\n\nWe investigate the applicability of a random-matrix model to the description of non-collective excitations in heavy-ion reactions around the Coulomb barrier. To this end, we study fusion in the reaction 16 O + 208 Pb, taking account of the known non-collective excitations in the 208 Pb nucleus. We show that the random-matrix model for the corresponding couplings reproduces reasonably well the exact calculations, obtained using empirical deformation parameters. This implies that the model may provide a powerful method for systems in which the non-collective couplings are not so well known.\n\n## I. INTRODUCTION\n\nHeavy-ion reactions around the Coulomb barrier often show a behavior that cannot be accounted for by a simple potential model [1][2][3]. They have thus provided a good opportunity to investigate the role of internal degrees of freedom in the reaction process. One of the well known examples is a large enhancement of sub-barrier fusion cross sections due to the couplings between the relative motion of the projectile and target nuclei and their internal degrees of freedom, such as surface vibrations for spherical nuclei, or rotational motion for nuclei possessing a static, intrinsic deformation. It is well recognized that these couplings lead to a distribution of potential barriers [4], and a method was proposed by Rowley, Satchler and Stelson to extract the barrier distributions directly from experimental fusion cross sections [5]. The barrier distributions extracted in this way are found to be sensitive to details of the couplings, often showing a characteristic structured behavior [1,6,7]. Similar heavy-ion barrier distributions can also be defined for large-angle quasi-elastic scattering [8,9].\n\nIn order to analyze experimental data for these lowenergy heavy-ion reactions, the coupled-channels method has been employed as a standard approach [3,10]. This method describes the reaction in terms of the internal excitations of the colliding nuclei, representing the total wave function of the system as a superposition of wave functions for the relevant reaction channels. Conventionally, a few low-lying collective excitations, that are strongly coupled to the ground state, are taken into account in these calculations. Such analyses have successfully accounted for the strong enhancement of subbarrier fusion cross sections, and have successfully reproduced the structure of the fusion and quasi-elastic barrier distributions for many systems [1,3].\n\nRecently, quasi-elastic barrier distributions have been measured for the 20 Ne + 90,92 Zr systems [11]. The corresponding coupled-channels calculations show that the main structure of these barrier distributions is determined by the rotational excitations of the strongly de-formed nucleus 20 Ne. The calculated barrier distributions are in fact almost identical for the two systems, even when the collective excitations of the Zr isotopes are taken into account. It was, therefore, surprising when the two experimental barrier distributions were found to be different in an important respect. That is, the barrier distribution for 20 Ne + 92 Zr exhibits a much more smeared behavior than that for the 20 Ne + 92 Zr system. The origin of this difference has been conjectured in Ref. [11] to be the multitude of non-collective excitations of the Zr isotopes, that are generally ignored in a coupled-channels analysis. In fact, the two extra neutrons in the 92 Zr nucleus lead to a significantly larger number of non-collective excited states compared with 90 Zr, since this latter possesses an N = 50 closed shell (the difference is reflected by the number of known states up to an excitation energy of 5 MeV; one finds 35 for 90 Zr and 87 for 92 Zr [12]).\n\nThere are many ways to describe non-collective excitations in heavy-ion reactions [13][14][15][16][17][18][19][20][21][22][23][24]. In the 1970's, Weidenmüller et al. introduced a random-matrix model for such excitations in order to study deep inelastic collisions [18][19][20][21][22][23][24]. In Ref. [13], we have used a similar model in a schematic one-dimensional barrier-penetration problem, in order to study the role of these non-collective excitations in low-energy reactions. On the other hand, in Ref. [14], we have explicitly taken into account in the coupled-channels formalism all of the 70 known noncollective states in 208 Pb below 7.382 MeV [25,26] without resorting to the random-matrix model, and have analysed in this way the experimental data for the 16 O+ 208 Pb reaction. Although some discrepancies between the experimental and theoretical barrier distributions remain after the inclusion of non-collective excitations, we have shown in Ref. [14] that these excitations play a more important role as the incident energy increases. We have also compared there the role of noncollective excitations in the fusion and quasi-elastic barrier distributions, and have shown that they affect both distributions in a similar fashion.\n\nGiven that exact calculations with a realistic spectrum for non-collective states is possible here, it is intriguing to also apply the random-matrix model to this system in order to test its applicability. This theoretical test is the main aim of this paper, and we achieve it by comparing our new results with those obtained in Ref. [14]. Note that, in contrast to 208 Pb, the properties of the noncollective states in 90,92 Zr are not known sufficiently well. As we will show in this paper, the random-matrix model provides a good method for a description of non-collective excitations in such a situation.\n\nThe paper is organized as follows. In Sec. II, we explain the coupled-channels formalism with non-collective excitations based on the random-matrix model. In Sec. III, we discuss the strength distribution and fusion cross sections obtained with this model. We then compare these with calculations using the more exact couplings and discuss the applicability of the random-matrix model. The paper is summarised in Sec. IV.\n\n## II. COUPLED-CHANNELS METHOD WITH NON-COLLECTIVE EXCITATIONS\n\nIn order to describe internal excitations during the reaction process, we assume the following Hamiltonian,\n\nwhere r is the separation of the projectile and target nuclei, and µ is the reduced mass. In this equation, H 0 ({ξ}) is the intrinsic Hamiltonian with {ξ} representing a set of internal degrees of freedom. The optical potential for the relative motion is V rel (r), and it includes an imaginary part to simulate the fusion process (that is, strong absorption into compound-nucleus degrees of freedom inside the Coulomb barrier). The coupling Hamiltonian between the relative motion and the intrinsic degrees of freedom is denoted by V coup (r, {ξ}).\n\nThe coupled-channels equations for this Hamiltonian are obtained by expanding the total wave function in terms of the eigenfunctions of H 0 ({ξ}). The equations read,\n\nwhere ǫ n is the excitation energy for the n-th channel.\n\nIn deriving these equations, we have employed the isocentrifugal approximation [3,[27][28][29][30][31][32]. In this approximation, the orbital angular momentum in the centrifugal potential is replaced by the total angular momentum J, thereby considerably reducing the dimension of the coupled-channels equations.\n\nIn solving these equations, we impose the following asymptotic boundary conditions,\n\nfor r → ∞, together with the regular boundary condition at the origin. Here, k n = 2µ(E -ǫ n )/ 2 is the wave number for the n-th channel, where n = 0 corresponds to the entrance channel. S J n is the nuclear S-matrix, and H (-) J (kr) and H (+) J (kr) are the incoming and the outgoing Coulomb wave functions, respectively. The fusion cross sections are then obtained as\n\nIn the random-matrix model [18][19][20][21][22][23][24], one assumes an ensemble of coupling matrix elements whose first moment satisfies\n\nwhile the second moment satisfies\n\nHere, I is the spin of the intrinsic state labeled by n, and α λ is the coupling form factor. In this paper, for simplicitly, we assume that the noncollective excitations couple only to the ground state, as in the linear coupling approximation employed in our previous work [14]. For the form factor α λ, we assume the following dependence\n\nwhere ρ(ǫ n ) is the level density at an excitation energy ǫ n, and (w λ, ∆, σ) are adjustable parameters. The appearance of the level density in the denominator reflects the complexity of the non-collective excited states, as discussed in Ref. [21]. For the function h(r), we adopt the derivative of the Woods-Saxon potential, that is,\n\nNote that this choice of the form factor corresponds to the coupling Hamiltonian in the linear coupling approximation derived from the Woods-Saxon potential. The histogram represents the experimental data [25], while the dashed line shows its fit with a polynomial function up to the sixth order.\n\n$$H = - 2 2µ ∇ 2 + V rel (r) + H 0 ({ξ}) + V coup (r, {ξ}),(1)$$\n\n$$- 2 2µ d 2 dr 2 + J(J + 1) 2 2µr 2 + V rel (r) + ǫ n -E u J n (r) + m V nm (r)u J m (r) = 0, (2$$\n\n$$)$$\n\n$$u J n (r) → H (-) J (k n r)δ n,0 - k 0 k n S J n H (+) J (k n r),(3)$$\n\n$$σ fus (E) = π k 2 0 J (2J + 1) 1 - n S J n 2. (4$$\n\n$$)$$\n\n$$V II ′ nn ′ (r) = 0,(5)$$\n\n$$V II ′ nn ′ (r)V I ′′ I ′′′ n ′′ n ′′′ (r ′ ) = {δ nn ′′ δ n ′ n ′′′ δ II ′′ δ I ′ I ′′′ + δ nn ′′′ δ n ′ n ′′ δ II ′′′ δ I ′ I ′′ } × (2I + 1)(2I ′ + 1) λ I λ I ′ 0 0 0 2 × α λ (n, n ′ ; I, I ′ ; r, r ′ ). (6$$\n\n$$)$$\n\n$$α λ (n, 0; I, 0; r, r ′ ) = w λ ρ(ǫ n ) e -ǫ 2 n 2∆ 2 e -(r-r ′ ) 2 2σ 2 h(r)h(r ′ ),(7)$$\n\n$$h(r) = e (r-R)/a 1 + e (r-R)/a 2. (8$$\n\n$$)$$\n\n## III. APPLICABILITY OF RANDOM-MATRIX MODEL A. Strength distribution\n\nLet us now apply the random-matrix model to the 16 O+ 208 Pb reaction and discuss its applicability. We first discuss the strength distribution for the non-collective excitations in 208 Pb obtained with the random-matrix model. To this end, we define the strength distribution as This quantity essentially corresponds to the \"square root\" of Eq. ( 6), except for an overall scale factor (here we have assumed w �� = w for all λ and omitted w λ in the definition for the strength function). The level density in Eq. ( 9) is treated in the following way. It is originally defined by\n\nfor a discrete spectrum. For practical purposes, we define the function\n\nthat gives the number of levels up to the excitation energy ǫ. We fit this function with a polynomial in ǫ, and then define a continuous level density by differentiating this polynomial. Figure 1 shows the experimental N (ǫ) for 208 Pb [25] in the interval between 4 MeV and 7.5 MeV (solid line) and its fit with a polynomial\n\na n ǫ n (dashed line). The values of a n are a 0 = -7479, a 1 = 6969 (MeV -1 ), a 2 = -2612 (MeV -2 ), a 3 = 497.5 (MeV -3 ), a 4 = -49.59 (MeV -4 ), a 5 = 2.347 (MeV -5 ), and a 6 = -0.03632 (MeV -6 ). The continuous level density, ρ(ǫ) = df (ǫ)/dǫ, is shown in Fig. 2. The strength distribution b I calculated with this level density is shown in Fig. 3 by the solid line as a function of excitation energy ǫ. The parameter ∆ in Eq. ( 9) is chosen to be 7 MeV, as in Refs. [23,24]. For comparison, the figure also shows the distribution of the experimental deformation parameters β I [25], smeared with a gaussian function with a width of 0.15 MeV (dashed line). We have also performed the same smearing for the strength distribution b I. Also, since the dimensions of β I and b I are not the same, the deformation parameters β I are scaled by a factor 10 so that the heights of the first peaks at about 4.3 MeV match one another. Although there exists a small deviation for the peaks between 5 MeV and 7 MeV, the overall structure of the strength distribution is well reproduced by this model.\n\n$$b I = λ 0 λ I 0 0 0 2 2I + 1 ρ(ǫ) e -ǫ 2 2∆ 2 = 2I + 1 ρ(ǫ) e -ǫ 2 2∆ 2.(9)$$\n\n$$ρ(ǫ) = n δ(ǫ -ǫ n ) (10$$\n\n$$)$$\n\n$$N (ǫ) = ǫ 0 ρ(ǫ ′ )dǫ ′ = n θ(ǫ -ǫ n ),(11)$$\n\n$$f (ǫ) = 6 n=0$$\n\n## B. Fusion cross sections\n\nThe strength distribution discussed in the previous subsection determines the coupling strength to each excited state. Let us then examine how the random-matrix model can be compared with the exact results in terms of the fusion cross sections for the 16 O + 208 Pb system. For this purpose, we use the same Woods-Saxon potential for the nuclear potential as in Ref. [14]; it has a surface diffuseness a = 0.671 fm, a radius R = 8.39 fm and a depth V 0 = 550 MeV. For the couplings to the collective excitations, we take into account the vibrational 3 - state at 2.615 MeV, the 5 -state at 3.198 MeV, and the 2 + state at 4.085 MeV in 208 Pb. The octupole mode is included up to the two-phonon states, while the other, weaker, vibrational modes are taken into account only up to their one-phonon states. The deformation parameters for these vibrational modes are estimated from the measured electromagnetic transition probabilities. They are β 3 = 0.122, β 5 = 0.058, and β 2 = 0.058 together with a radius parameter of r 0 =1.2 fm. Although we took into account the octupole phonon state of 16 O in our preivous study [14], for simplicity we do not include it in the present calculations, since its effect can be well described by an adiabatic renormalization of the potential depth [3,33]. For the parameter σ in Eq. ( 7), we follow Refs. [23,24] and use σ = 4 fm. On the other hand, the parameter w λ = w is chosen to be w = 38000 MeV 3/2 so that the height of the main peak in the fusion barrier distribution is reproduced by the random-matrix model.\n\nFigures 4 (a) and 4 (b) show the 16 O+ 208 Pb fusion excitation function on linear and logarithmic scales respectively. The dashed lines show the results obtained with the measured deformation parameters for the noncollective excitations, while the solid lines show the results obtained using the random-matrix approximation. For comparison, the dotted lines show results that account only for the collective excitations. Although a small overall shift can be seen, it is clear that the randommatrix model reproduces the exact results reasonably well.\n\nIn order to highlight the energy dependence, Fig. ergy dependence of fusion cross sections due to the noncollective excitations is similar in the two calculations. In particular, both barrier distributions are smeared out in a similar way at energies around 80 MeV, and both calculations yield a similar second peak around 87.5 MeV. (We note that if the strength w 0 was somewhat larger, the second peak could appear at even higher energies, possibly reflecting the broad bump seen at around 97 MeV in the experimental data.)\n\nAs we have argued in Ref. [13], the higher-energy peaks in the barrier distribution are affected more by noncollective excitations than are the lower-energy peaks.\n\nUnfortunately this is not easy to see in Fig. 4 because the peaks obtained with purely collective couplings are not resolved. This difference can, however, be easily understood using perturbation theory. That is, the eigenchannels corresponding to the higher-energy peaks in the barrier distribution couple more strongly to the noncollective states via their ground state component simply because the energy differences are smaller. Higher peaks are thus redistributed more, effectively removing much of their strength from that region of energy.\n\nFrom these calculations, it is evident that the effects of non-collective excitations are not sensitive to details of the non-collective couplings, and that the random-matrix model is applicable to the description of non-collective excitations, so long as the relevant parameters are chosen appropriately.\n\n## IV. SUMMARY\n\nWe have investigated the applicability of the randommatrix model for the description of non-collective excitations in low-energy heavy-ion reactions. To this end, we have calculated the fusion excitation function for the 16 O + 208 Pb system, where the role of the non-collective excitations has already been investigated in our previous study using empirical deformation parameters.\n\nWe have first shown that the coupling strength distribution obtained with the random-matrix model agrees well with the experimental distribution. The fusion cross section and barrier distribution for the 16 O + 208 Pb sys-tem obtained with empirical non-collective couplings are also well reproduced by the random-matrix model with appropriately chosen parameters. These results provide a validation of the random-matrix model for the description of non-collective couplings.\n\nFor the 208 Pb nucleus, detailed properties of noncollective states are known over a large energy range. However, this is not always the case for other systems. That is, for many nuclei, even though the energies and spin-parity may be relatively well known for many noncollective states, the coupling strengths are poorly determined. In such a situation, the present study suggests that the random matrix model provides a powerful tool to treat these coupling strengths. A good example is the quasi-elastic barrier distribution for the 20 Ne + 90,92 Zr systems, where it has been suggested that non-collective excitations may play an important role. Analyses for these systems within the random-matrix model are under way. We shall report the results in a separate publication [34].\n\n## References\n\n1. Dasgupta, Hinde, Rowley et al. (1998) *Annu. Rev. Nucl. Part. Sci*\n\n2. Balantekin, Takigawa (1998) *Rev. Mod. Phys*\n\n3. Hagino, Takigawa (2012) *Prog. Theor. Phys*\n\n4. Dasso, Landowne, Winther (1983) *Nucl. Phys. A*\n\n5. (1983) *A*\n\n6. Rowley, Satchler, Stelson (1991) *Phys. Lett. B*\n\n7. Leigh (1995) *Phys. Rev. C*\n\n8. Leigh, Rowley, Lemmon et al. (1993) *Phys. Rev. C*\n\n9. Timmers, Leigh, Dasgupta et al. (1995) *Nucl. Phys. A*\n\n10. Hagino, Rowley (2004) *Phys. Rev.C*\n\n11. Hagino, Rowley, Kruppa (1999) *Compt. Phys. Comm*\n\n12. Piasecki, Świderski, Gawlikowicz et al. (2009) *Phys. Rev. C*\n\n13. \"Evaluated Nuclear Structure Data File\"\n\n14. Yusa, Hagino, Rowley (2010) *Phys. Rev. C*\n\n15. Yusa, Hagino, Rowley (2012) *Phys. Rev. C*\n\n16. Diaz-Torres, Hinde, Dasgupta et al. (2008) *Phys. Rev. C*\n\n17. Diaz-Torres (2010) *Phys. Rev. C*\n\n18. Diaz-Torres (2010) *Phys. Rev. C*\n\n19. Ko, Pirner, Weidenmüller (1976) *Phys. Lett*\n\n20. Agassi, Weidenmüller, Ko (1978) *Phys. Lett*\n\n21. Barrett, Shlomo, Weidenmüller (1978) *Phys. Rev. C*\n\n22. Agassi, Ko, Weidenmüller (1977) *Ann. Phys*\n\n23. Ko, Agassi, Weidenmüller (1979) *Ann. Phys*\n\n24. Agassi, Ko, Weidenmüller (1979) *Ann. Phys*\n\n25. Agassi, Ko, Weidenmüller (1978) *Phys. Rev. C*\n\n26. Wagner, Crawley, Hammerstein et al. (1975) *Phys. Rev. C*\n\n27. Lewis, Bertrand, Fulmer (1973) *Phys. Rev. C*\n\n28. Lindsay, Rowley (1984) *J. Phys. G*\n\n29. Nagarajan, Rowley, Lindsay (1986) *J. Phys. G*\n\n30. Nagarajan, Balantekin, Takigawa (1986) *Phys. Rev. C*\n\n31. Esbensen, Landowne, Price (1987) *Phys. Rev. C*\n\n32. (1987) *C*\n\n33. Tanimura (1987) *Phys. Rev. C*\n\n34. (1987) *Z. Phys. A*\n\n35. Gomez-Camacho, Andres, Nagarajan (1994) *Nucl. Phys. A*\n\n36. Takigawa, Hagino, Abe et al. (1994) *Phys. Rev. C*\n\n37. Yusa, Hagino, Rowley<|endoftext|>"
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+      "example": "# Comparison of results from a 2+1D relativistic viscous hydrodynamic model to elliptic and hexadecapole flow of charged hadrons measured in Au-Au collisions at √ s NN = 200 GeV\n\nVictor Roy, A Chaudhuri, Bedangadas Mohanty\n\n## Abstract\n\nSimulated results from a 2+1D relativistic viscous hydrodynamic model have been compared to the experimental data on the centrality dependence of invariant yield, elliptic flow (v2), and hexadecapole flow (v4) as a function of transverse momentum (pT ) of charged hadrons in Au-Au collisions at √ sNN = 200 GeV. Results from two types of initial transverse energy density profile, one based on the Glauber model and other based on Color-Glass-Condensate (CGC) are presented.We observe no difference in the simulated results on the invariant yield of charged hadrons for the calculations with different initial conditions. The comparison to the experimental data on invariant yield of charged hadrons supports a shear viscosity to entropy density ratio (η/s) between 0 to 0.12 for the 0-10% to 40-50% collision centralities. The simulated v2(pT ) is found to be higher for a fluid with CGC based initial condition compared to Glauber based initial condition for a given collision centrality. Consequently the Glauber based calculations when compared to the experimental data requires a lower value of η/s relative to CGC based calculations. In addition, a centrality dependence of the estimated η/s is observed from the v2(pT ) study. The v4(pT ) for the collision centralities 0-10% to 40-50% supports a η/s value between 0 -0.08 for a CGC based initial condition. While simulated results using the Glauber based initial condition for the ideal fluid evolution under estimates the v4(pT ) for collision centralities 0-10% to 30-40%.\n\n## I. INTRODUCTION\n\nHeavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) have provided evidence for the formation of a hot and dense QCD matter [1][2][3][4][5]. This presents an unique opportunity to study the transport properties, like shear viscosity to entropy density ratio (η/s), of the QCD matter. There are two main theoretical approaches to estimate the value of η/s from the experimental data. One based on a microscopic approach as in transport theory [6][7][8][9][10] and other related to a macroscopic approach through relativistic viscous hydrodynamic calculations [11][12][13][14][15][16][17][18]. In this work, we will compare our results from a relativistic 2+1 dimension viscous hydrodynamics to recent high statistics experimental data on elliptic (v 2 ) and hexadecapole (v 4 ) flow of charged hadrons measured by the PHENIX Collaboration [19,20]. The experimental observables related to azimuthal anisotropic flow are found to be sensitive to shear viscous effects. The shear viscosity decreases the anisotropy of the fluid velocity. Hence v 2 and v 4 as a function of transverse momentum (p T ) are expected to decrease with the increase in the value of η/s.\n\nOne of the main uncertainties in the estimation of η/s using a viscous hydrodynamics simulation is due to the choice of the initial conditions [14,17,21]. In this work we have considered two models, Glauber and Color Glass Condensate (CGC), to obtain the initial transverse energy density profile. For this study we have considered a smooth initial condition, which does not vary event-by-event. Previous work have shown that both the spatial and momentum anisotropy are expected to be larger for a CGC based initial condition compared to Glauber model [17]. Hence for other similar conditions in the simulations, the calculations with CGC based initial condition is expected to give higher values of v 2 compared to initialization based on a Glauber model. Earlier comparisons of viscous hydrodynamic simulations with both CGC and Glauber initial conditions to the experimental data at RHIC can be found in Refs [17,18,21,22]. In Ref [17,18], the experimental data used for comparison are the centrality dependence of multiplicity, p T, p T integrated v 2, and minimum bias v 2 vs. p T for charged hadrons in Au-Au collisions at √ s N N = 200 GeV. In general it was observed that calculations with CGC based initial condition prefers a higher value of η/s compared to calculations with a Glauber based initial condition. In Ref [21,22] the authors have tried to explain the centrality dependence of v 2 divided by the eccentricity with a viscous hydrodynamic model for the QGP phase coupled to a transport model for the hadronic phase. Comparison of the experimental data to the calculations done for CGC initial condition supports a η/s value ∼ 0.16 -0.24. While the corresponding comparisons for a Glauber model based initial condition supports a lower value of η/s ∼ 0.08 -0.16.\n\nIn the current work we compare the results from the viscous hydrodynamics simulations with two different initial conditions (Glauber and CGC) to recent high statistics measurements of v 2 (p T ) and v 4 (p T ) of charged hadrons in Au-Au collisions at √ s N N = 200 GeV for a broad range in collision centrality (from 0-10% to 40-50%) [19]. We also compare the simulated results to the measured charged particle invariant yields as a function of p T for various collision centralities [20].\n\nThe paper is organized as follows. In the next section we discuss the formalism of viscous hydrodynamic model used in this work. This includes a brief discussion on the energy-momentum conservation and relaxation equations for shear stress. We present a detailed discussion on the initial conditions used in the calculations. The equation of state used and the freeze-out conditions are also presented. In section III we present a comparative study between calculations with Glauber and CGC initial conditions of various observables in the simulation. These includes the temporal evolution of shear stress, average transverse velocity, and eccentricity. Section IV presents the comparison of viscous hydrodynamic simulations with different input values of η/s for both Glauber and CGC based initial conditions to the experimental data on invariant yield versus p T, v 2 (p T ), and v 4 (p T ) for various collision centralities. Finally in section V we present a summary of the work.\n\n## II. VISCOUS HYDRODYNAMIC SIMULATION\n\nIn a relativistic viscous hydrodynamics scenario, there are two-fold corrections to the ideal fluid hydrodynamics. In presence of the dissipative processes, the energy momentum tensor contains additional dissipative corrections. The equilibrium freezeout distribution function used in the Cooper-Frey freezeout prescription [23] also gets modified. The first order dissipative correction to the energy-momentum tensor leads to acausal behavior [24]. The second order causal viscous hydrodynamics due to Israel-Stewart is one of the most commonly used theory [25]. For the simulation results presented here, we will follow the Israel-Stewart formalism for the evolution of a viscous fluid using the 2+1D viscous hydrodynamic code \"AZHYDRO-KOLKATA\" [26,27]. Shear viscosity is the only dissipative process considered in our present study. We assume a net-baryon free plasma is formed in Au-Au collisions at midrapidity at √ s N N = 200 GeV.\n\n## A. Conservation and relaxation equations\n\nThe energy-momentum conservation equation and relaxation equation for shear viscosity in Israel-Stewart formalism is expressed as\n\nEquation 1 is the conservation equation for the energy-momentum tensor, T µν = (ε + p)u µ u νpg µν + π µν. ε, p, and u are the energy density, pressure, and fluid velocity respectively. π µν is the shear stress tensor. Equation 2 is the relaxation equation for the\n\n) is a symmetric traceless tensor. η is the shear viscosity and τ π is the corresponding relaxation time. Assuming longitudinal boost-invariance, the above equations are solved with the code 'AZHYDRO-KOLKATA' in (τ = √ t 2z 2, x, y, η s = 1 2 ln t+z t-z ) coordinates. Where τ is the longitudinal proper time, (t, x, y, z) are spacetime coordinates, and η s is the space time rapidity.\n\n$$∂ µ T µν = 0,(1)$$\n\n$$Dπ µν = - 1 τ π (π µν -2η∇ <µ u ν> ) -[u µ π νλ + u ν π µλ ]Du λ. (2$$\n\n$$)$$\n\n$$π µν. D = u µ ∂ µ is the convec- tive time derivative, ∇ <µ u ν> = 1 2 (∇ µ u ν + ∇ ν u µ ) - 1 3 (∂ µ u µ )(g µν -u µ u ν$$\n\n## B. Initial conditions\n\nThe initial conditions used here includes the initial energy density profile in the transverse plane (ǫ(x, y)), the initial time (τ i ), the transverse velocity profile (v x (x, y), v y (x, y)), shear stresses in the transverse plane (π µν (x, y)) at τ i. The τ i value is taken as 0.6 fm. The η/s values are also inputs to the viscous hydrodynamics simulations. We have taken the following temperature independent values for this work, η/s = 0, 0.08, 0.12, 0.16, and 0.18.\n\nWe have considered two different models for the calculation of initial energy density profile in the transverse plane. One is based on a two component Glauber model. At an impact parameter b, the transverse energy density is obtained from the following two component form\n\nwhere N part (b, x, y) and N coll (b, x, y) are the transverse profile of participant numbers and binary collision numbers respectively. ǫ 0 corresponds to the central energy density in b = 0 and does not depend on the impact parameter of the collision. The parameter x h is the hard scattering fraction. Both ǫ 0 and x h are fixed to reproduce the experimental charged hadron multiplicity density at midrapidity. The N part (b, x, y) and N coll (b, x, y) values are obtained using an optical Glauber model calculation [28]. The value of x h is found to be 0.9 and the values of ǫ 0 for various input values of η/s are given in the Table I. The other model commonly used to obtain initial conditions for hydrodynamics is the Color-Glass-Condensate (CGC) approach, based on the ideas of gluon saturation at high energies [29,30]. We have used the KLN (Kharzeev-Levin-Nardi) k Tfactorization approach [31], due to Drescher et al. [32].\n\nWe follow references [17,33] and consider that the initial energy density can be obtained from the gluon number density through the thermodynamic relation,\n\nwhere dNg d 2 xT dY is the gluon number density evaluated at central rapidity Y = 0 and the overall normalization C is a free parameter. C is fixed to reproduce the experimentally measured charged particle multiplicity density at midrapidity. The values of C used in the simulations for different input values of η/s are given in Table I. The number density of gluons produced in a collision of two nuclei with mass number A is given by\n\nwhere p T and Y are the transverse momentum and rapidity of the produced gluons, respectively.\n\nx 1,2 = p T × exp(±Y )/ √ s is the momentum fraction of the colliding gluon ladders with √ s the center of mass collision energy and α s (k T ) is the strong coupling constant at momentum scale k T ≡ |k T |. N is the normalization constant. The unintegrated gluon distribution functions are taken as\n\nP (x T ) is the probability of finding at least one nucleon at transverse position x T and is defined as\n\n, where T A is the thickness function and σ is the nucleon-nucleon cross section taken as 42 mb. The saturation scale at a given momentum fraction x and transverse coordinate x T is given by Q\n\nThe growth speed is taken to be λ = 0.28. Shear stresses π µν is initialized to their corresponding Navier-Stokes estimates for the boost invariance velocity profile, π xx = π yy = 2η/3τ i, π xy = 0 [34]. We have used τ π = 3η/4p (where η and p are the shear viscous coefficient and pressure) in our simulation, which corresponds to the kinetic theory estimates of relaxation time for shear viscous stress for a relativistic Boltzmann gas [25]. The initial values of v x (x, y) and v y (x, y) are taken to be zero.\n\n$$ǫ(b, x, y) = ǫ 0 [(1-x h ) N part 2 (b, x, y)+x h N coll (b, x, y)](3)$$\n\n$$ǫ(τ i, x T, b) = C × dN g d 2 x T dY (x T, b) 4/3,(4)$$\n\n$$dN g d 2 x T dY = N d 2 p T p 2 T pT d 2 k T α s (k T ) φ A (x 1, (p T + k T ) 2 /4; x T ) φ A (x 2, (p T -k T ) 2 /4; x T ),(5)$$\n\n$$φ(x, k 2 T ; x T ) = 1 α s (Q 2 s ) Q 2 s max(Q 2 s, k 2 T ) P (x T )(1 -x) 4,(6)$$\n\n$$P (x T ) = 1 -1 -σTA A A$$\n\n$$2 s (x, x T ) = 2 GeV 2 TA(xT )/P (xT ) 1.53/fm 2 0.01 x λ.$$\n\n## C. Equation of state\n\nIn the present simulations we have used an equation of state with cross-over transition at temperature T c = 175 MeV [35]. The low temperature phase of the EoS is modeled by hadronic resonance gas, containing all the resonances with M res ≤2.5 GeV. The high temperature phase is a parametrization of the recent lattice QCD calculation [36]. Entropy density of the two phases are joined at T = T c = 175 MeV by a smooth step like function. The thermodynamic variables pressure (p), energy density (ε), entropy density (s) etc. are then calculated by using the standard thermodynamic relations\n\n$$p (T ) = T 0 s (T ′ ) dT ′ (7) ε (T ) = T s (T ) -p (T ).(8)$$\n\n## D. Freeze-out condition\n\nThe hydrodynamic expansion of the hot and dense matter leads to cooling of the system. After some time the mean-free path of the constituent becomes large/comparable to the system size. The system can no longer maintain the local thermal equilibrium and the momentum distribution of the particles remains unchanged after that. This is called freezeout. We use the Cooper-Frey algorithm at the freezeout to calculate invariant yields of the hadrons [23]. The freezeout temperature which is a free parameter in the hydrodynamics simulation is taken as T f =130 MeV. The effect of different choices of freeze-out temperature on charged hadron p T spectra and elliptic flow is discussed in appendix A.\n\nAs we have already pointed out there are twofold correction to the ideal fluid in the presence of viscous effects. The freezeout distribution function for a system slightly away from local thermal equilibrium can be approximated as [26]\n\nwhere φ(x, p) << 1 is the corresponding deviation from the equilibrium distribution function f eq (x, p). The non-equilibrium correction φ(x, p) can be approximated in Grad's 14 moment method by a quadratic function of the four momentum p µ in the following way [37,38]\n\nwhere ε, ε µ, and ε µν are functions of p µ, metric tensor g µν, and thermodynamic variables.\n\nFor our study where only shear stresses are considered, φ(x, p) has the following form\n\nwhere\n\nAs expected, the correction factor increases with increasing values of shear stress π µν at freezeout. The correction term also depends on the particle momentum. The Cooper-Frey formula [23] for a non equilibrium system is\n\nwhere g is the degeneracy of the particle considered and dΣ µ is the normal to the elemental freeze-out hypersurface.\n\n$$f neq (x, p) = f eq (x, p)[1 + φ(x, p)],(9)$$\n\n$$φ(x, p) = ε -ε µ p µ + ε µν p µ p ν,(10)$$\n\n$$φ(x, p) = ε µν p µ p ν,(11)$$\n\n$$ε µν = 1 2(ǫ + p)T 2 π µν. (12$$\n\n$$)$$\n\n$$dN d 2 p T dy = g (2π) 3 dΣ µ p µ f neq (p µ u µ, T ),$$\n\n## III. GLAUBER VERSUS CGC INITIAL CONDITION\n\nA. Space-time evolution\n\nFigure 1 shows the constant temperature contours corresponding to T c = 175 MeV and T f = 130 MeV in the τ -x plane (at y = 0) indicating the boundaries for the QGP and hadronic phases respectively. The results are from the viscous hydrodynamic simulations for Au-Au collisions at impact parameter 7.4 fm and η/s = 0.08. The solid red curves corresponds to initial transverse energy density profile based on CGC model and the dashed black curve corresponds to results based on Glauber model initial conditions. We observe that the lifetime of QGP and hadronic phases are slightly larger for the simulations based on CGC initial conditions compared to Glauber based initial conditions. The spatial extent of the hadronic phase is slightly smaller for the simulations with CGC initial conditions relative to Glauber based conditions.\n\n## B. Temporal evolution of shear stress\n\nIn presence of shear viscosity the thermodynamic pressure is modified. The tracelessness of shear stress tensor π µν, along with the assumption of longitudinal boost invariance ensures that at the initial time π xx and π yy components of shear viscous stress are positive. Consequently in viscous fluid the effective pressure in the transverse direction is larger compared to the ideal fluid, for the same thermodynamic condition. It is then important to have some idea how various components of shear viscous stress π µν evolves in space-time. We have considered π xx, π yy, and π xy as the three independent components of shear stress π µν. This choice is not unique.\n\nThe temporal evolution of spatially averaged π xx, π yy, and π xy are shown in Fig. 2 for CGC (solid red curve) and Glauber (black dashed curve) initialization of energy density. All the three components of π µν becomes zero after time ∼ 7 fm irrespective of the CGC or Glauber model initialization. At initial time the values of spatially averaged π xx and π yy are observed to be larger for CGC compared to the Glauber initialization. However, the difference vanishes quickly ∼ 3 fm. For π xy a noticeable difference is seen for CGC and Glauber model initialization within time ∼ 6 fm., where\n\n. The angular bracket... implies an energy density weighted average. Solid red curve is for CGC based initial condition and the dashed black curve is for the Glauber based initial condition. We observe almost no change in the v T as a function of time for the two initial conditions studied. This effect should be reflected in the slope of the invariant yield of the charged hadrons as a function of transverse momentum being same for both the initial conditions. These results are discussed in section IV A.\n\nFigure 4 shows the temporal evolution of the spatial eccentricity (ε x ) and the momentum space anisotropy (ε p ) of the viscous fluid (η/s = 0.08) with Glauber and CGC based initial conditions for Au-Au collisions at impact parameter, b = 7.4 fm. The ε x which is a measure of the spatial deformation of the fireball from spherical shape is defined as and the ε p which is a measure of the asymmetry of fireball in momentum space is defined as\n\nwhere T xx and T yy are the components of energymomentum tensor T µν. Solid red curve is for CGC based initial condition and the dashed black curve is for the Glauber based initial condition. We find both ε x and ε p are higher for the simulated results with CGC based initial condition compared to initial condition based on Glauber model. As the simulated elliptic flow v 2 in hydrodynamic model is directly related to the temporal evolution of the momentum anisotropy, we expect the v 2 for the CGC based initial condition to be larger than the corresponding values for the Glauber based initial condition. These results are discussed in section IV B.\n\n$$γ = 1 √ 1-v 2 x -v 2 y$$\n\n$$ε x = y 2 -x 2 y 2 + x 2,(13)$$\n\n$$ε p = dxdy(T xx -T yy ) dxdy(T xx + T yy ),(14)$$\n\n## IV. COMPARISON TO EXPERIMENTAL DATA\n\nThe experimental data used for comparison to our simulated results are from the PHENIX collaboration at RHIC [19,20]. The observables used are invariant yield of charged hadrons, elliptic flow, and hexadecapole flow as a function of p T for Au-Au collisions at pseudorapidity | η |< 0.35 for √ s NN = 200 GeV. The high statistics recent PHENIX measurement of elliptic (k = 1) and hexadecapole (k = 2) flow [19] are obtained using the formula v 2k = cos(2k(φ -Ψ 2 )) after correction of the event plane resolution. Where φ is the azimuthal angle of the charged hadrons and Ψ 2 is the second order event plane constructed using event plane detectors in 1.0 <| η |< 3.9. The rapidity gap between the detectors used to measure the v 2k and Ψ 2 ensures absence of significant ∆η dependent non-flow correlations, which are also absent in our hydrodynamic simulations. Ψ 2 for our simulation is along x axis. We compare below our simulated results on invariant yield, elliptic, and hexadecapole flow for five different collision centralities with input η/s varying between 0.0 to 0.18 to the corresponding experimental data.\n\nA. Invariant yield laboration [20]. The simulated results are from the 2+1D relativistic viscous hydrodynamic model with a Glauber based initial transverse energy density profile. The black solid, orange long dashed, purple dash-dotted, magenta short dashed and green dotted lines corresponds to calculations with η/s = 0.0, 0.08, 0.12, 0.16, and 0.18 respectively. We find the 0-10% experimental data is best explained by simulation with η/s = 0.0. Whereas data for collision centralities between 20-30% to 40-50% supports a η/s value within 0.08 to 0.12. Figure 6 shows the same results as in Fig. 5 but the simulated results corresponds to 2+1D viscous hydrodynamic calculations with a CGC based initial transverse energy density profile. The conclusions regarding the comparison between simulated results and experimental data are similar to that obtained for Fig. 5. This also means that the invariant yield of charged hadrons are not very sensitive to the choice of a Glauber based or CGC based initial conditions. The average transverse velocity at the freeze-out which determines the slope of the p T spectra was observed to be similar for the fluid evolution with Glauber and CGC based initial conditions (see Fig. 3).\n\n## B. Elliptic flow\n\nFigure 7 shows the elliptic flow (v 2 ) as a function of transverse momentum (p T ) for charged hadrons at midrapidity in Au-Au collisions at √ s NN = 200 GeV. The results are shown for five different collision centralities (0-10%, 10-20%, 20-30%, 30-40%, and 40-50%). The open circles are the experimental data from the PHENIX collaboration [19]. The simulated results are from the 2+1D relativistic viscous hydrodynamic model with a Glauber based initial transverse energy density profile. The black solid, orange long dashed, purple dash-dotted, magenta short dashed, and green dotted lines corresponds to calculations with η/s = 0.0, 0.08, 0.12, 0.16, and 0.18 respectively. We find the experimental data prefers higher values of η/s as we go from central to peripheral collisions. While 0-10% collision centrality experimental data is best described by ideal fluid (η/s = 0.0) simulation results, those corresponding to 40-50% collision centrality is closest to simulated results with η/s = 0.18. Figure 8 shows the same results as in Fig. 7 but the simulated results corresponds to 2+1D viscous hydrodynamic calculations with a CGC based initial transverse energy density profile. Also shown for comparison the simulated results for ideal fluid evolution with Glauber based initial conditions. We find the v 2 (p T ) for CGC based initial condition is larger compared to corresponding results from Glauber based initial conditions. This can be understood from the fact that CGC based initial condition leads to a higher value of momentum anisotropy compared to Glauber based initial condition (as seen in Fig. 4). The general conclusion that the experimental data prefers a higher value of η/s as we go from central to peripheral collisions as seen for viscous hydrodynamic simulations with Glauber based initial conditions also holds for those with the CGC based initial conditions. However, we find from the comparison of experimental data to simulations based on CGC initial conditions that the v 2 (p T ) data for 0-10% collisions is best explained for simulated results with η/s between 0.08-0.12. This is in contrast to what we saw from the comparisons of data to simulations with Glauber based initial conditions, where the data preferred η/s = 0.0 (see Fig. 7). For more peripheral collisions (centralities beyond 20-30%), it seems data would prefer a higher value of η/s ∼ 0.18. We do not present simulation results for η/s > 0.18 as the viscous hydrodynamic simulated spectra distributions show a large deviations from ideal fluid simulation results (see appendix B). This leads to a breakdown of the simulation frame work which is designed to be valid for case of small deviations of observables from ideal fluid simulations.\n\n## C. Hexadecapole flow\n\nFigure 9 shows the hexadecapole flow (v 4 ) as a function of transverse momentum (p T ) for charged hadrons at midrapidity in Au-Au collisions at √ s NN = 200 GeV. The results are shown for five different collision centralities (0-10%, 10-20%, 20-30%, 30-40%, and 40-50%). The open circles are the experimental data from the PHENIX collaboration [19]. Simulated results for only ideal fluid evolution using Glauber based initial condition are shown (solid black curve). While for the CGC based initial conditions the simulated results are shown for η/s = 0.0 (purple solid thick curve) and 0.08 (orange dashed curve). We do not present simulated v 4 results for other η/s values as these are much lower compared to the data. We find that v 4 (p T ) from ideal hydrodynamic simulations with Glauber based initial conditions under predict the experimental data for all collision centralities studied except for the most peripheral collisions (40-50%) presented. This is in sharp contrast to the observation for v 2 (p T ) (see Fig. 7) under similar conditions. Comparsion between simulated results with CGC based initial condition and experimental data shows that the preferred η/s lies between 0.0 and 0.08 for the collision centralities studied. The η/s values supported by the data on v 2 and v 4 using the simulated results presented here appears to be different. In this study we have used smooth initial conditions, a more realistic approach is to use a fluctuating initial condition and carry out event-by-event hydrodynamics. This will enable us to study the odd flow harmonics v 3 along with the even harmonics v 2 and v 4. The simultaneous description of all these experimentally measured flow harmonics in a viscous hydrodynamics framework will probably provide a better estimation of η/s.\n\n## V. SUMMARY\n\nWe have carried out a 2+1D relativistic viscous hydrodynamic simulation with two different initial conditions (Glauber and CGC) for the transverse energy density profile in Au-Au collisions at √ s N N = 200 GeV. The simulations are carried out for η/s values between 0.0 to 0.18, using a lattice + hadron resonance gas model based equation of state which has a cross over temperature for the quark-hadron transition at 175 MeV. The shear viscous corrections are considered both in the evolution equations and freeze-out distribution function. We find that the temporal dependence of the average transverse velocity of the viscous fluid is similar for both the initial conditions studied. The components of shear viscous stress are observed to have higher values for the simulations with CGC initial conditions compared to those for Glauber model initialization at early times of fluid evolution (< 6 fm). The simulated invariant yield of charged particles as a function of transverse momentum is also found to be similar for the Glauber and CGC based initial conditions. The spatial eccentricity and the momentum anisotropy have larger values for simulations with CGC based initial condition compared to the corresponding values for Glauber based initial condition. The simulated elliptic flow is observed to be higher for calculations with CGC based initial con-ditions relative to those with Glauber based initial conditions, for a given collision centrality.\n\nWe have compared our simulated results to the experimental data at midrapidity on the centrality dependence of invariant yield, v 2, and v 4 as a function of p T of charged hadrons measured in Au-Au collisions at √ s NN = 200 GeV. From the comparison to the p T spectra of charged particles we observe that the data supports a η/s value between 0 to 0.12 for the 0-10% to 40-50% collision centralities for both the initial conditions considered. The v 2 (p T ) experimental data requires a lower value of η/s for simulations with Glauber model initialization compared to the CGC based initial conditions. For both the models of initial conditions the v 2 (p T ) data indicates a centrality dependence in the estimated η/s value, with peripheral collisions preferring larger values. The experimental data on v 4 (p T ) for the collision centralities 0-10% to 40-50% supports a η/s value between 0 -0.08 for a CGC based initial condition. While simulated results using the Glauber based initial condition for the ideal fluid evolution under estimates the v 4 (p T ) for collision centralities 0-10% to 30-40%. Simulations with Glauber model initial conditions explain the v 4 (p T ) data for 40-50% collisions with η/s = 0.0. The observation associated with v 4 is different from v 2, with v 4 data preferring smaller values of η/s.\n\nThere are further scopes of improvement on the simulated results presented here. Recent experimental measurements of odd and higher order azimuthal anisotropic flow [39][40][41] suggests that a fluctuating initial condition needs to be considered. It is expected that the input η/s to the hydrodynamic simulations has a temperature dependence in both the QGP and hadronic phases [11]. Although large uncertainties still exist in the QCD computations of η/s for the QGP phase. A more precise estimation of η/s would require the viscous fluid simulations to also consider bulk viscosity and vorticity effects [42]. Both of which are expected to be non-zero for the system formed in high energy heavy-ion collisions and may affect the observables like v 2. A proper prescription for bulk viscous freeze-out correction is still under debate in literature [27,43,44], while implementation of vorticity in viscous hydrodynamic simulations have just started to be investigated. We plan to consider some of these effects in the near future. We have studied the effect of different freeze-out temperature on the charged hadron invariant yield and elliptic flow. Figure 10 shows the invariant yield (panel -a) and elliptic flow (panel -b) of charged hadrons for 20-30% centrality Au-Au collisions at √ s NN = 200 GeV. All the simulated results are for η/s = 0.08 using the same Glauber based initial condition but with three different freeze-out temperatures, T f = 110 (red dashed curve),130 (black solid curve), and 150 MeV (blue dash-dotted curve).\n\nThe slope of the p T spectra increases as T f decreases. This is because of higher radial velocity gained due to longer duration of evolution of the system for the lower freeze-out temperature. The experimentally measured p T spectra is best explained for simulations with input parameters as specified in section II and T f = 130 MeV.\n\nFor the current simulations the effect of different T f values studied is observed to be small on elliptic flow of charged hadrons. This could possibly be due to saturation of the value of the momentum anisotropy at the early time of evolution.\n\n## Appendix B: Viscous correction\n\nThere are two kinds of dissipative correction to the ideal fluid simulation. First the energy momen-tum tensor contains a viscous correction and the freezeout distribution function is also modified in presence of the dissipative processes. The viscous hydrodynamics model is applicable when the dissipative correction (both in the energy-momentum tensor and freezeout distribution function) is small compared to the corresponding equilibrium value. It is then implied that the relative viscous correction (δN/N eq ) is small for the Israel-Stewart's hydrodynamics to be applicable, where N eq is the invariant yield for system in local thermal equilibrium, and δN = dN d 2 pT | viscous -dN d 2 pT | equilibrium. In figure 11 the relative viscous correction δN/N eq for η/s = 0.18 are shown for 0-10% (black solid curve) and 40-50% (black dashed curve) collision centrality. We observe that the relative shear viscous correction is quite large (∼ 50%) at p T ∼1.0 GeV for 40-50% centrality (dashed curve in figure 11). The solid curve in the same figure shows the relative viscous correction to the invariant yield of charged hadron for Au-Au collision for 0-10% centrality at √ s NN = 200 GeV. A higher value of η/s will introduce a larger viscous correction and eventually the viscous hydrodynamics framework will no longer be applicable.\n\n## References\n\n1. Arsene (2005) *Nucl. Phys. A*\n\n2. Back (2005) *Nucl. Phys. A*\n\n3. Adams (2005) *Nucl. Phys. A*\n\n4. Adcox (2005) *Nucl. Phys. A*\n\n5. Gyulassy, Mclerran (2005) *Nucl. Phys. A*\n\n6. Xu, Ko (2011) *Phys. Rev. C*\n\n7. Greco (2011) *J. Phys. Conf. Ser*\n\n8. Alver, Gombeaud, Luzum et al. (2010) *Phys. Rev. C*\n\n9. Demir, Bass (2009) *Eur. Phys. J. C*\n\n10. Bhalerao, Blaizot, Borghini et al. (2005) *Phys. Lett. B*\n\n11. Niemi, Denicol, Huovinen et al. (2011) *Phys. Rev. Lett*\n\n12. Shen, Heinz, Huovinen et al. (2010) *Phys. Rev. C*\n\n13. Heinz, Moreland, Song (2009) *Phys. Rev. C*\n\n14. Chaudhuri (2009) *Phys. Lett. B*\n\n15. Bozek (2010) *Phys. Rev. C*\n\n16. Schenke, Jeon, Gale (2011) *Phys. Rev. Lett*\n\n17. Luzum, Romatschke (2008) *Phys. Rev. C*\n\n18. Luzum, Romatschke (2009) *Phys. Rev. Lett*\n\n19. Adare (2010) *Phys. Rev. Lett*\n\n20. Adler (2004) *Phys. Rev. C*\n\n21. Song, Bass, Heinz et al. (2011) *Phys. Rev. Lett*\n\n22. Song, Bass, Heinz et al. (2011) *Phys. Rev. C*\n\n23. Cooper, Frye (1974) *Phys. Rev. D*\n\n24. Hiscock, Lindblom (1985) *Phys. Rev. D*\n\n25. Israel, Stewart (1979) *Annals Phys*\n\n26. (1976) *Ann. Phys. (N.Y.)*\n\n27. Chaudhuri\n\n28. Roy, Chaudhuri (2012) *Phys. Rev. C*\n\n29. Miller, Reygers, Sanders et al. (2007) *Ann. Rev. Nucl. Part. Sci*\n\n30. Mclerran, Venugopalan (1994) *Phys. Rev. D*\n\n31. Mclerran, Venugopalan (1994) *Phys. Rev. D*\n\n32. Kharzeev, Levin, Nardi (2004) *Nucl. Phys. A*\n\n33. Drescher, Dumitru, Hayashigaki et al. (2006) *Phys. Rev. C*\n\n34. Dumitru, Molnar, Nara (2007) *Phys. Rev. C*\n\n35. Chaudhuri (2009) *Phys. Lett. B*\n\n36. Roy, Chaudhuri (2011) *Phys. Lett. B*\n\n37. Borsanyi (2010) *JHEP*\n\n38. Muronga, Rischke\n\n39. Muronga (2007) *Phys. Rev. C*\n\n40. Adare (2011) *Phys. Rev. Lett*\n\n41. Aamodt (2011) *Phys. Rev. Lett*\n\n42. Aad\n\n43. Becattini, Piccinini, Rizzo (2008) *Phys. Rev. C*\n\n44. Monnai, Hirano (2009) *Phys. Rev. C*\n\n45. Dusling, Schafer (2012) *Phys. Rev. C*<|endoftext|>"
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+      "example": "# On the maximal displacement of critical branching random walk in random environment\n\nWenxin Fu, Wenming Hong\n\n## Abstract\n\nIn this article, we study the maximal displacement of critical branching random walk in random environment. Let M n be the maximal displacement of a particle in generation n, and Z n be the total population in generation n, M be the rightmost point ever reached by the branching random walk. Under some reasonable conditions, we prove a conditional limit theorem,where random variable A Λ is related to the standard Brownian meander. And there exist some positive constant C 1 and C 2, such thatCompared with the constant environment case (Lalley and Shao ( 2015)), it revaels that, the conditional limit speed for M n in random environment (i.e., n) is significantly greater than that of constant environment case (i.e., n 1 2 ), and so is the tail probability for the M (i.e., x -2 3 vs x -2 ). Our method is based on the path large deviation for the reduced critical branching random walk in random environment.\n\n## 1 Introduction\n\nSpatial branching systems have been extensively investigated over past decades, in which the maxima of the n-th generation of the branching random walk is one of the keynotes. For supercritical branching random walk (m > 1), the law of large numbers for the maxima of branching random walk in generation n, M n, can trace back to Hammersley [19], Kingman [24], Biggins [8] and Bramson [13]. Since then extensively studied on this topics have appeared in recent years, see for example [1,6,7,11,12,21] and references therein. In particular Aïdékon proved in [6] that the distribution of the centered maximal converges in law to a random shift of the Gumbel distribution (see also [11]).\n\nFor critical cases, the system will die out eventually, the maximal displacement of the system is finite almost surely, and it is natural to consider the tail probability of the maximal displacement. The asymptotic law for the maxima of a critical branching Brownian motion trace back to Sawyer and Fleischman [32] and Lalley and Sellke [26]. While for critical branching random walk, results appeared in recent years. The case when the offspring distribution is critical, that is m = 1, was considered by Kesten [23] and Lalley and Shao in [27]. Let M n be the maximal displacement in generation n, and Z n be the total population in generation n. By introducing the discrete Feynman-Kac formula, Lalley and Shao proved in [27], under some moment assumptions (Theorem 3 in [27]),\n\nwhere G is a nontrivial distribution that depends only on the variances of the offspring and step distributions. For M, the rightmost point ever reached by the branching random walk, (Theorem 1 in [27]),\n\nHere α is a constant which depends on the standard deviations of the jump and offspring distribution.\n\nConsider a branching random walk in random environment (BRWre), i.e., a branching random walk with the time-inhomogeneous environment, which has been introduced by Biggins and Kyprianou in [9]. For the maximal displacement in generation n, M n, of the supercritical branching random walk in random environment, Huang and Liu [22] proved that the maximal displacement in the process grows at ballistic speed almost surely. Mallein and Mi loś investigated the second order behavior in [31], i.e.,\n\nin probability with respect to the annealed law (i.e., averaging over the branching random walk and point process laws). Here, θ * and ϕ are deterministic constants for which descriptions are given in [31], and K n is an environment-measurable random walk. The random environment make effects on both the speed of M n, that is the limit (in probability) of M n, is strictly greater than that seen in the time-homogeneous case and the logarithmic correction is also strictly greater than in the time-homogeneous case (see [31]).\n\nIn this article, we focus on the critical branching random walk in random environment. Different from the method \" discrete Feynman-Kac formula\" by Lalley and Shao in [27], we apply the path large deviation method to prove (Theorem 1), for M n, conditional on survival events (i.e. {Z n > 0}), the annealed law of the maximal displacement in generation n convergence weakly to some non-degenerated random variable A Λ, i.e.\n\n$$L M n n 1 2 |Z n > 0 =⇒ G, as n → ∞,(1)$$\n\n$$P (M x) ∼ α x 2, as x → ∞. (2$$\n\n$$)$$\n\n$$lim n→∞ M n -Kn θ * log n = -ϕ(3)$$\n\n## L\n\nM n √ σn\n\nFor M, the rightmost position ever reached by the branching random walk, we prove that (Theorem 2), P(M x) ≍ α x 2/3, as x → ∞.\n\n(\n\nunder the annealed probability P.\n\nTo formulate precisely, let P (N 0 ) be the space of probability measures on N 0 := {0, 1, • • • }. Equipped with the metric of total variation, P (N 0 ) becomes a Polish space. For any probability measure F ∈ P (N 0 ), we also use F to denote the generating function of this probability measure on N, the mean value and normalized second factorial moment of F is denoted as\n\nAlso, denote by κ(F, a) the standardized truncated second moment of the probability measure F ∈ P (N 0 ), κ(F, a) := 1\n\nLet F be a random variable taking values in P (N 0 ). Then an infinite sequence ξ := {F 1, F 2, • • • } of i.i.d. copies of F is said to form a random environment, we use P denote the corresponding probability of environment.\n\nGiven an environment ξ := {F 1, F 2, • • • }, the time-inhomogeneous branching random walk in environment ξ is a process constructed as follows:\n\n• It starts with one individual located at the origin in time 0.\n\n• At time n (n 1), each individual alive at time n -1 dies and reproduces several children according to the probability measure F n.\n\n• Every new child moves independently according to some jump distribution µ respect to its parent (and independent of all other factors).\n\nWe denote by T the (random) genealogical tree of the process. For a given individual u ∈ T, we use V (u) ∈ R for the position of u and |u| for the generation where u is alive. The pair (T, V ) is called the branching random walk in the time-inhomogeneous environment ξ. For each n 0, define Z n := #{u ∈ T, |u| = n} the number of particles survived in generation n.\n\nGiven the environment ξ, use P ξ denote the quenched probability of the branching random walk (T, V ). And use P = P ⊗ P ξ denote the annealed probability of the branching random walk in random environment. Define\n\nUnder P, {X k ; k 1} is a sequence of i.i.d copies of logarithmic of the mean of the offspring number X := log F, and the sequence S := (S 0, S 1, • • • ) is called the associated random walk.\n\n$$3 4 |Z n > 0 =⇒ L (A Λ ), as n → ∞.(4)$$\n\n$$)5$$\n\n$$F := ∞ k=0 kF [k], F := 1 F 2 ∞ k=1 k(k -1)F [k].(6)$$\n\n$$F 2 ∞ y=a y 2 F [y].$$\n\n$$X k := log F k, η k := Fk, S 0 = 0, and S n = S n-1 + X n.(7)$$\n\n## Assumption 1. Assume,\n\n(1) For the offspring\n\nand there exist positive integer a and some positive number δ > 0, such that\n\nHere log + x := log (max{x, 1}).\n\n(2) The jump distribution µ is standard normal distribution N (0, 1).\n\n$$E[X] = 0, σ 2 := E[X 2 ] ∈ (0, ∞).$$\n\n$$E log + κ(F, a) 2+δ < ∞.$$\n\n## Remark 1.\n\n(1) Under the assumption, the branching processes in random environment will extinct eventually, actually, there exist some positive and finite constant K, such that\n\nThis was first proved by Kozlov ([25]) for linear fractional critical branching processes in i.i.d. environments, and then, Afanasyev et.al ( [5]) extended to more general cases as for Assumption 1.\n\n(2)For simplicity, the jump distribution we consider is the standard normal distribution N (0, 1). In fact, it is reasonable for any distribution that satisfies the Cramér's condition such that we can deal with the path large deviation for the branching random walk. This is the usually condition for the supercritical branching random walk. Actually, conditioned on non-extinction, we will consider the reduced critical branching random walk in random environment which behaves more like time-inhomogeneous supercritical case.\n\nAccording to eq. ( 8), the branching system will die out eventually, that means the maximal displacement of the branching system is finite almost surely. In this article, we will investigate the maximal displacement in generation n condition on survival events (i.e. {Z n > 0}), and the whole maximal displacement of this system (T, X), i.e.\n\nConditioned on survival events {Z n > 0}, we will consider the maximal displacement in generation n (i.e. M n ) firstly, and obtain the following conditional limit theorem.\n\nTheorem 1. Under the Assumption 1, condition on the survival events {Z n > 0}, M n √ σn\n\nconvergence in law to some non-degenerated random variable A Λ :\n\nHere\n\n) is the set of continuous functions on interval [0, 1] with value 0 at 0. For the whole maximal displacement (i.e. M ) of critical branching random walk in random environment, we prove the asymptotic order of the tail probability. Compared with the behavior of M n in eq. ( 1) and eq. ( 9), it reveals that in the critical case, the conditional limit speed for M n in random environment (i.e., n 3 4 ) is significantly greater than that of constant environment ( time-homogeneous) case (i.e., n 1 2 ), and so is the tail probability for the M (see eq. ( 2) and eq. ( 10)). The reason behind the phenomena is that, by Yaglom theorem for the critical branching processes, conditioned on non-extinction, the number of the particles Z n in generation n is Z n ∼ n for the constant environment and Z n ∼ e cn 1/2 for the random environment case. As a consequence, the significantly larger number of living particles (in the random environment) makes it possible to jump to a higher position even though we assume the step jump is light tail (satisfying the Cramér's condition condition, see Assumption 1). Actually, conditioned on non-extinction, we need only to consider the reduced critical branching random walk in random environment which behaves more similar to the time-inhomogeneous supercritical case, which enable us to prove the conditional limit by the path large deviation and then get the tail behavior for the M, the rightmost point ever reached by the branching random walk.\n\nThe rest of the paper is as follows. In Section 2, we introduce a kind of time-inhomogeneous branching random walk. And we will establish the sample path deviation principle Theorem 3 under Assumption 2 in Section 2.1. Based on the deviation principle, we will investigate the maximal displacement in generation n (i.e. Theorem 10) under Assumption 2 and Assumption 3, which will play an important role in proving Theorem 1.\n\nIn Section 3, we consider the reduced critical branching processes in random environment, and proved a stronger conditional limit theorem, which ensures the conditional reduced critical branching random walk in random environment satisfies Assumption 2 and Assumption 3 in Section 2, thus Theorem 10 can be applied.\n\nFinally, with the tools established in Section 2 and Section 3 in hand, we will prove Theorem 1 in Section 4.1 and prove Theorem 2 in Section 4.2.\n\n$$lim n→∞ √ nP (Z n > 0) = K. (8$$\n\n$$)$$\n\n$$M n := max{X(u) : u ∈ T, |u| = n}, M := sup M n.$$\n\n$$3 4$$\n\n$$L M n √ σn 3 4 |Z n > 0 =⇒ L(A Λ ), as n → ∞.(9)$$\n\n$$A Λ := sup{g(1) : ∀r ∈ [0, 1], ´r 0 1 2 g ′ (s) 2 ds Λ r, g ∈ C 0 ([0, 1])}, where Λ t := inf{W + s ; s ∈ [t, 1]} is the minimal process of a standard Brownian meander {W + t ; t ∈ [0, 1]} and C 0 ([0, 1]$$\n\n## 2 Time-inhomogeneous Branching Random Walk\n\nFor the time-inhomogeneous branching random walk, in [15], Fang and Zeitouni study the maximal displacement of branching random walks in a class of time inhomogeneous environments. In the main results, an interesting phenomenon appears about the asymptotic behavior of the maximum displacement: the profile of the variance matters, both to the leading (velocity) term and to the logarithmic correction term, and the latter exhibits a phase transition. In [29], Mallein consider more general time-inhomogeneous branching random walk, and obtain similar results that is related to a optimization problem.\n\nIn [16], Fang and Zeitouni investigate a kind of time-inhomogeneous branching Brownian motion, obtain the leading (velocity) term and specify the correction term being of order n 1 3 ; later in [28], Maillard and Zeitouni refined the results: set m t = v(1)tw(1)n 1 3 σ(1) log n and some suitable m * t satisfies sup |m * tm t | < ∞, where w(1) is related to the largest zero of the Airy function of the first kind, the tightness about M tm t, the convergence in law about {M tm * t } and the limit distribution was obtained. And about time-inhomogeneous branching random walk, Mallein proved similar results in [30].\n\nIn present paper, we deal with the M n, the maximal displacement in generation n of the branching random walk in random environment. It is reasonable to realize that only the particles alive in generation n make contributions to M n. So conditioned on non-extinction, we need only to consider the so called reduced critical branching random walk in random environment, which behaves more like time-inhomogeneous supercritical case.\n\nTo this end, in this section we will introduce a more general framework of the time-inhomogeneous branching random walk and establish the sample path moderate deviation principle, and then prove the so-called the law of large number of the maximal displacement M n of time-inhomogeneous branching random walk under some conditions.\n\n## 2.1 Model and Assumptions\n\nFor each n,\n\n) n is a n length sequence of branching mechanism. Given F n, we can construct a time-inhomogeneous branching random walk up to time n, whose jump distribution is N (0, 1) and independent of the branching, we use (T n, V n ) denote this branching system. N n k denote the set of particles alive at time k, and Z n k := |N n k | be the number of particles lives in generation k. P n and E n denote the probability and the expectation of the time-inhomogeneous branching random walk. Assumption 2. For each n and 1 k n, we assume that\n\nAssume there exist a sequence finite positive number {a n } such that lim\n\nAssume f (t) being some continuous non-decreasing function on [0, 1], satisfies f (0) = 0 and\n\nRemark 2. The asymptotic relationship satisfied by the sequence {a n } appeared in Assumption 2 is consistent with the moderate deviation of the random walk with Cramér's condition, see Theorem 3.7.1 in [14]. And for the normal distribution cases, the condition lim n→∞ an n = 0 can be omitted.\n\nProof of Lemma 1. Define the continuous path\n\nUnder Assumption 2, it is not hard to find the increasing functions f n (t) also convergence to f (t) in [0, 1]. For any ǫ > 0, due to the continuous of f, we know there exist δ > 0, such for\n\n), so there exist C(k) large enough such that for any\n\nLet C := max{C(k) : 0 k N }, then for any n > C and all x ∈ [0, 1),\n\nThus the proof is over.\n\nAssumption 3. Under Assumption 2, for the time-inhomogeneous branching processes, for any ε > 0, and each t ∈ (0, 1],\n\nRemark 3. About the eq. ( 11) in Assumption 2, it is a technical assumption. What's more, if all the branching mechanisms belong to the linear fractional case, eq. ( 11) can result to the Assumption 3 directly (just calculate the generating function). However, for more general cases, although Assumption 2 are satisfied, Assumption 3 may be false.\n\nFor classical supercritical branching processes (non-extinction), when the L log L condition is satisfied, the martingale {W n := Zn m n } uniformly convergence to some non-negative random variable W, and satisfies P(W > 0) = P ({survival forever}) = 1 which means the Assumption 3 is right. However, for general time-inhomogeneous branching process, we don't have such results. In Section 2.6, when estimating the maximal displacement of branching random walk, Assumption 3 plays an essential role. Under Assumption 3, the time-inhomogeneous branching processes evolve very close to the supercritical cases, thus we can use the deviation results together with exponent increasing property to estimate the lower bound of M n n.\n\n$$F n := {F n 1, F n 2, • • •, F n n } ∈ P (N 0$$\n\n$$F n k [0] = 0, then the mean of F n k satisfies F n k 1. Let S n 0 := 0 and S n k := k i=1 X n i for 1 k n, where X n i := log F n i 0.$$\n\n$$F n k = 0. (11$$\n\n$$)$$\n\n$$f (t) > 0 for t > 0. For any t ∈ [0, 1], lim n→∞ S n ⌊nt⌋ a n = f (t).$$\n\n$$Lemma 1. Under Assumption 2, S n ⌊n•⌋ a n ; n 1 converges to f (•) uniformly.$$\n\n$$f n (t) :=      S n ⌊nt⌋ a n + 1 a n (nt -⌊nt⌋) X ⌊nt⌋+1, t ∈ [0, 1) S n n a n, t = 1.$$\n\n$$any |x -y| < δ, then |f (x) -f (y)| < ǫ. Choose some N large enough, such 1 N < ǫ, then for any 0 k N -1, f ( k N ) -f ( k + 1 N ) < ǫ. For each 0 k N, as f n ( k N ) → f ( k N$$\n\n$$n > C(k), f n ( k N ) -f ( k N ) < ǫ.$$\n\n$$|f n (x) -f (x)| f (x) -f n ( ⌊N x⌋ N ) + f n ( ⌊N x⌋ N ) -f ( ⌊N x⌋ N ) + f ( ⌊N x⌋ N ) -f (x) < f n ( ⌊N x⌋ + 1 N ) -f n ( ⌊N x⌋ N ) + ǫ + ǫ f ( ⌊N x⌋ + 1 N ) + ǫ -f ( ⌊N x⌋ N ) -ǫ + 2ǫ = f ( ⌊N x⌋ + 1 N ) -f ( ⌊N x⌋ N ) + 4ǫ < 5ǫ.$$\n\n$$lim n→∞ P n log Z n ⌊nt⌋ a n -f (t) > ε = 0$$\n\n## 2.2 The Sample Path Moderate Deviation Principle\n\nFor any ν ∈ N n n, we define following path re-scale.\n\nFor any r ∈ [0, 1], the space C 0 ([0, r]) is the set of continue functions on [0, r] with value 0 at 0, and endow it with the supremum norm •. For any r > 0, set H r := {g(t); g ′ ∈ L 2 [0, r]} denote the space of all absolutely continuous function with value 0 at 0 that posses a square integrable derivation. Use I r (•) denote the Schilder rate function of Brownian motion on C 0 ([0, r]):\n\nAccording to the Schilder theorem, I r is a good rate function: for any 0 α < ∞, the level set\n\nFor the path of time-inhomogeneous branching random walk, we have following results, which can be viewed as a kind of moderate deviation principle. Theorem 3. Under Assumption 2, there is a moderate deviation principle for the path of the time-inhomogeneous branching random walk, i.e.,\n\nHere S f (g\n\nRemark 4. Hardy and Harris prove related results for binary branching Brownian motion in [20], and we just extend such path deviation principle to the general time-inhomogeneous branching system.\n\nAbout the rate function S f (•) in Theorem 3, we have\n\n. And due to I 1 (•) is a good rate function, we know {g : I 1 (g) α + f (1)} is a compact set. So, for any subsequence g n k such that g n k convergence to some function g, we know g also satisfies\n\nFor each r ∈ [0, 1], and I r is also a good rate function, confined to the interval [0, r], we have\n\nDue to the arbitrariness of r ∈ [0, 1], then S f (g) α. Due to the arbitrariness of subsequence, we know {g : S f (g) α} is a compact set. The proof of Theorem 3 is mainly divided into three parts: in Section 2.3, depending on the many-to-one formula, we will prove the local upper bound; in Section 2.4, using the spine decomposition, we will prove the local lower bound; in Section 2.5, follow the steps of Section 7 in [20], firstly establish the weakly deviation principle and due to the goodness of rate function which means the exponential tightness, then Theorem 3 will be proved.\n\n$$Definition 1. For any ν ∈ N n n, for each 0 k n, use V n ν (k) denote the position of the k-generation ancestor of ν, thus the path of ν is V n ν := {V n ν (0), V n ν (1), • • •, V n ν (n)}. Define the function V n,an ν on [0, 1] to be the re-scaled path of ν, V n,an ν (t) :=      1 √ na n ((⌊nt⌋ + 1 -nt) V n ν (⌊nt⌋) + (nt -⌊nt⌋) V n ν (⌊nt⌋ + 1)), t ∈ [0, 1). 1 √ na n V n ν (n), t = 1.$$\n\n$$I r (g) := 1 2 ´r 0 g ′ (s) 2 ds g ∈ H r +∞ otherwise.$$\n\n$${f ∈ C 0 ([0, r]) : I r (f ) α} is a compact subset of C 0 ([0, r]).$$\n\n$$• Upper bound: If C is a closed subset of C 0 ([0, 1]), then lim sup n→∞ 1 a n log P n (∃ν ∈ N n n : V n,an ν ∈ C) -inf g∈C S f (g). • Lower bound: If G is an open subset of C 0 ([0, 1]), then lim inf n→∞ 1 a n log P n (∃ν ∈ N n n : V n,an ν ∈ G) -inf g∈G S f (g).$$\n\n$$) :=      sup r∈[0,1] ´r 0 1 2 g ′ (s) 2 ds -f (r), for g ∈ H 1 ; +∞, otherwise.$$\n\n$$Theorem 4. For any α ∈ [0, ∞), the level set {g : S f (g) α} is compact in C 0 ([0, 1]), which means S f (g) is a good rate function on C 0 ([0, 1]). Proof of Theorem 4. Fixed α ∈ [0, ∞), for any g n ∈ C 0 ([0, 1]) such that S f (g n ) α, then I 1 (g n ) α + f (1)$$\n\n$$I 1 (g) α + f (1) < ∞.$$\n\n$$ˆr 0 1 2 g ′ (s) 2 ds -f (r) = I r (g) -f (r) lim inf k→∞ I r (g n k ) -f (r) = lim inf k→∞ ˆr 0 1 2 g ′ n k (s) 2 ds -f (r) lim inf k→∞ S f (g n k ) α.$$\n\n## 2.3 Local Upper Bound\n\nWe at first need the path moderate deviation principle of the random walk.\n\n) be a random walk with one step distribution N (0, 1), denote µ n as the law of\n\nin C 0 ([0, 1]), then {µ n } satisfies the Large Deviation Principle with the good rate function\n\nProof of Lemma 2. We can enlarge the probability space, let {B t, t ∈ [0, ∞)} is a standard Brownian motion. Note that µ n is the law of B an n (•) in C 0 ([0, 1]). Use ν n denote the law of\n\n), due to the scale property of Brownian motion, ν n is also the\n\nAccording to the Schilder Theorem (see Theorem 5.2.3 in [14]), we know ν n satisfied the LDP with rate function I 1 (•) at speed a n in C 0 ([0, 1]).\n\nAnd for each ǫ > 0,\n\nHere under P 0,0 1,0, {B t, t ∈ [0, 1]} is a Brownian bride form 0 at time 0 to 0 at time 1, and the last equality is based on the the probability of a Brownian bridge to stay below a straight line. And it follows, by considering n → ∞, that\n\nTherefore, the probability measure µ n and ν n are exponentially equivalent. By Theorem 4.2.13 in [14], the proof of Lemma 2 now is over.\n\nRemark 6. In fact, for normal distribution N (0, 1), the condition lim n→∞ an n = 0 can be omitted in Lemma 2. And for more general random walk which satisfies the Cramér's condition, together with the moderate deviation (see Theorem 3.7.1 in [14], and the condition lim n→∞ an n = 0 is important), we can prove Lemma 2 by repeating the proof of Theorem 5.1.2 in [14]. Now we can prove the local upper bound of Theorem 3.\n\nTheorem 5. For any g ∈ C 0 ([0, 1]), we have\n\nHere B δ (g\n\nProof of Theorem 5. First note that as δ → 0, eq. ( 12) is decreasing, which means the limit does exist. Under P n, the sequence\n\n) is a random walk with one step distribution N (0, 1). For each r ∈ [0, 1], Using many-to-one formula (in fact, calculate the mean of branching random walk),\n\nHere B an n (•) is defined in Lemma 2, and according to Lemma 2, we know\n\nTogether with eq. ( 13) and Assumption 2, then lim\n\nDue to the arbitrary of r ∈ [0, 1], the proof of Theorem 5 is over.\n\n$$Lemma 2. Assume a n → ∞ and an n → 0, let (B 0 := 0, B 1, • • •, B n$$\n\n$$B an n (•) :=      1 √ na n (⌊nt⌋ + 1 -nt) B ⌊nt⌋ + (nt -⌊nt⌋) B ⌊nt⌋+1, t ∈ [0, 1); 1 √ na n B n, t = 1.$$\n\n$$I 1 (•) at speed a n in C 0 ([0, 1]).$$\n\n$$B n (t) := 1 √ na n B nt in C 0 ([0, 1]$$\n\n$$law of B t √ a n in C 0 ([0, 1]).$$\n\n$$P(∃t ∈ [0, 1], B nt √ na n -B an n (t) > ǫ) n k=1 P (∃t ∈ [k -1, k], |(B t -B k ) -t(B k+1 -B k )| > ǫ √ na n ) =nP 0,0 1,0 max t∈[0,1] |B t | > ǫ √ na n nP 0,0 1,0 max t∈[0,1] B t > ǫ √ na n + nP 0,0 1,0 min t∈[0,1] B t < -ǫ √ na n =2nP 0,0 1,0 max t∈[0,1] B t > ǫ √ na n = 2n exp -2ǫ 2 na n$$\n\n$$lim n→∞ 1 a n log P(∃t ∈ [0, 1], B nt √ na n -B an n (t) > ǫ) = -∞.$$\n\n$$lim δ→0 lim sup n→∞ 1 a n log P n (∃ν ∈ N n n : V n,an ν ∈ B δ (g)) -S f (g). (12$$\n\n$$)$$\n\n$$) := {ρ ∈ C 0 ([0, 1]) : |ρ(t) -g(t)| < δ, ∀t ∈ [0, 1]}.$$\n\n$$(B 0, B 1, • • •, B n$$\n\n$$P n (∃ν ∈ N n n : V n,an ν ∈ B δ (g)) P n (∃ν ∈ N n n : |V n,an ν (s) -g(s)| < δ, ∀s ∈ [0, r]) P n ∃ν ∈ N n ⌊nr⌋ : |V n,an ν (s) -g(s)| < δ, ∀s ∈ [0, r] E n   ν∈N n ⌊nr⌋ ½ {|V n,an ν (s) -g(s)| < δ, ∀s ∈ [0, r]}   = exp S n ⌊nr⌋ P n (|B an n (s) -g(s)| < δ, ∀s ∈ [0, r])(13)$$\n\n$$lim δ→0 lim sup n→∞ 1 a n log P n (|B an n (s) -g(s)| < δ, ∀s ∈ [0, r]) - ˆr 0 1 2 g ′ (s) 2 ds.$$\n\n$$δ→0 lim sup n→∞ 1 a n log P n (∃ν ∈ N n n : V n,an ν ∈ B δ (g)) - ˆr 0 1 2 g ′ (s) 2 ds -f (r).$$\n\n## 2.4 Local lower bound\n\nThe following theorem describe the lower bound of Theorem 3.\n\nIn order to prove Theorem 6, we will change the measure to obtain a realization of target event. And in branching system, it is a very mature approach to do such by additive martingale.\n\nChoose a function g ∈ H 1, then ´1 0 1 2 g ′ (s) 2 ds < ∞. Consider the scale-up of g(t), for each 0 k n, let\n\nand define W n 0 := 0, and for 1 k n,\n\nUnder P n, the sequence {W n 0, W n 1, • • •, W n n } construct the so-called additive martingale. Thus, we can construct a new measure Q n which satisfies dQ n dP n F n k := W n k, here the filtration {F n k : 0 k n} is the natural filtration of the n-th branching random walk. What's more, under Q n, the branching system is a branching process with the spine and can be described as follow:\n\n• Initially, single particle stays at 0, and this particle is the 0-generation spine particle, denote it as ω n 0.\n\n• At generation k, the spine particle ω n k-1 dies, and gives birth to children according to the law {q n k [s] :=\n\n, then uniformly choose one as the k-generation spine particle, denote it as ω n k, other particles are normal. Besides the spine particle ω n k-1, other normal particles die and reproduce just according to F n k at generation k, and all these reproduced particles are normal.\n\n• The spine particle ω n k move according to N (λ n k, 1) respect to its parent ω n k-1 ; all normal particles move according to N (0, 1) respect to their own parent.\n\n} is just a random walk with one step distribution N (0, 1). About P n and Q n, we have\n\nNow we calculate the α-moment of W n n under Q n.\n\nLemma 3. Under Assumption 2, for any α ∈ [0, 1], and for any δ > 0, there exist N (δ) such that for all n > N (δ),\n\nProof of Lemma 3. Let the filtration G ∞ contains all information about the spine, and therefore we can obtain the spine decomposition:\n\nHere N (ω n k-1 ) represent the number of children that ω n k-1 has. Recall the following well-known facts.\n\nF n k, by Jeason's inequality, eq. ( 16), Remark 7 and Proposition 7, we have\n\nAccording to Lemma 1, for any δ > 0, there exist N such that for any n > N, 1\n\nWhat's more, for any 0 k n, use Cauchy-Schwarz Inequality,\n\nThen according to eqs. ( 17) to (19) and the definition of S f (g),\n\nThus, the proof is over.\n\nUsing Lemma 3, we have Theorem 8. Under Assumption 2, for any ǫ > 0,\n\nProof of Theorem 8. For each α ∈ (0, 1), (W n k ) α is a Q n submartingale (indeed, (W n k ) 1+α is P n submartingale), then according to Doob's submartingale inequality,\n\nAccording to Lemma 3, together with eq. ( 20),\n\nAccording to Assumption 2, due to the continuity of f, we know log max\n\nNow choose α small enough, such that α ´1 0 1 2 g ′ (s) 2 dsǫ < 0, then fix some δ small enough, which satisfies δ < α ǫα ´1 0 1 2 g ′ (s)ds. In eq. ( 21), take n → ∞, then\n\nThe proof is finished. Now, we are going to prove Theorem 6:\n\nProof of Theorem 6. According to eq. ( 14),\n\nAccording to Remark 7 and the law of large number, we know\n\nTogether with Theorem 8 and eq. ( 22),\n\nDue to the arbitrary of ǫ, the proof of Theorem 6 is over.\n\n$$Theorem 6. For g ∈ C 0 ([0, 1]), lim inf n→∞ 1 a n log P n (∃ν ∈ N n n : V n,an ν ∈ B δ (g)) -S f (g).$$\n\n$$g n,an (k) := √ na n g k n,and$$\n\n$$λ n i = g n,an (i) -g n,an (i -1), 1 i n.$$\n\n$$W n k := ν∈N n k exp(-S n k ) exp k i=1 λ n i [X n (ν(i)) -X n (ν(i -1))] - 1 2 (λ n i ) 2$$\n\n$$sF n k [s] F n k ; s = 1, 2, • • • } (the size-biased distribution of F n k )$$\n\n$$Remark 7. Under Q n, define V n (ω n k ) := V n (ω n k )-g n,an (k), then { V n (ω n 0 ), V n (ω n 1 ), • • •, V n (ω n n )$$\n\n$$P n (∃ν ∈ N n n : V n,an ν ∈ B δ (g)) = Q n 1 W n n ; ∃ν ∈ N n n : V n,an ν ∈ B δ (g)(14)$$\n\n$$Q n ((W n n ) α ) exp(a n δ) exp (αa n S f (g)) exp 1 2 α 2 a n ˆ1 0 g ′ (s) 2 ds 1 + n k=1 F n k F n k (15$$\n\n$$)$$\n\n$$Q (W n n |G ∞ ) = exp (-S n n ) exp n i=1 λ n i V n (ω n i ) -V n (ω n i-1 ) - 1 2 (λ n i ) 2 + n k=1 exp (-S n k ) N (ω n k-1 ) -1 exp k-1 i=1 λ n i V n (ω n i ) -V n (ω n i-1 ) - 1 2 (λ n i ) 2(16)$$\n\n$$Proposition 7. If α ∈ [0, 1] and u, v > 0 then (u + v) α u α + v α. Note that Q n N (ω n k-1 ) -1 = F n k$$\n\n$$Q n ((W n n ) α ) = Q n (Q n ((W n n ) α | G ∞ )) Q n (Q n (W n n |G ∞ ) α ) Q n exp (-αS n n ) exp n i=1 αλ n i ( V n (ω n i ) -V n (ω n i-1 )) + 1 2 α(λ n i ) 2 + n k=1 Q n exp(-αS n k ) N (ω n k-1 ) -1 exp k-1 i=1 αλ n i ( V n (ξ i ) -V n (ξ i-1 )) + 1 2 α(λ n i ) 2 = exp (-αS n n ) exp α 2 + α 2 n i=1 (λ n i ) 2 + n k=1 exp(-αS n k )F n k F n k exp α 2 + α 2 k-1 i=1 (λ n i ) 2(17)$$\n\n$$a n S n k -f ( k n ) < δ α, then αS n k αa n f ( k n ) -a n δ(18)$$\n\n$$k i=1 (λ n i ) 2 = na n k i=1 g( i n ) -g( i -1 n ) 2 a n ˆk n 0 g ′ (s) 2 ds. (19$$\n\n$$)$$\n\n$$Q n ((W n n ) α ) exp (a n δ -αa n f (1)) exp α 2 + α 2 a n ˆ1 0 g ′ (s) 2 ds + n k=1 exp a n δ -αa n f ( k -1 n ) F n k F n k exp α 2 + α 2 a n ˆk-1 n 0 g ′ (s) 2 ds exp(a n δ) exp (αa n S f (g)) exp α 2 a n ˆ1 0 1 2 g ′ (s) 2 ds 1 + n k=1 F n k F n k$$\n\n$$lim n→∞ Q n sup 0 k n W n k exp (a n (S f (g) + ǫ)) = 1.$$\n\n$$Q n max 0 k n W n k > exp (a n (S f (g) + ǫ)) = Q n max 0 k n (W n k ) α > exp (a n α(S f (g) + ǫ)) Q n ((W n n ) α ) exp (a n α(S f (g) + ǫ))(20)$$\n\n$$Q n max 0 k n W n k > exp (a n (S f (g) + ǫ)) exp(a n δ -a n ǫ) exp 1 2 α 2 a n ˆ1 0 g ′ (s) 2 ds 1 + n k=1 F n k F n k(21)$$\n\n$$1 k n {F n k } = o(a n ), thus log 1 + n k=1 F n k F n k log 1 + n k=1 F n k + log max 1 k n {F n k } = o(a n ).$$\n\n$$lim n→∞ Q n max 0 k n W n k > exp (a n (S f (g) + ǫ)) = 0.$$\n\n$$P n (∃ν ∈ N n n : V n,an ν ∈ B δ (g)) = Q n 1 W n n ; ∃ν ∈ N n n : V n,an ν ∈ B δ (g) exp(-a n (S f (g) + ǫ))Q n max 0 k n W n k exp (a n (S f (g) + ǫ)) ; ∃ν ∈ N n n, V n,an ν ∈ B δ (g) exp(-a n (S f (g) + ǫ))Q n max 0 k n W n k exp (a n (S f (g) + ǫ)) ; ω n n ∈ N n n, V n,an ω n n ∈ B δ (g).(22)$$\n\n$$lim n→∞ Q n ω n n ∈ N n n, V n,an ω n n ∈ B δ (g) = 1.$$\n\n$$lim inf n→∞ 1 a n log P n (∃ν ∈ N n n : V n,an ν ∈ B δ (g)) -S f (g) -ǫ.$$\n\n## 2.5 Improving the \"weak\" deviation result\n\nIn this subsection, we will prove Theorem 3. As the same structure in [20], repeat the Section 7 in [20], we can prove the weak deviation results, i.e. the upper bound in Theorem 3 holds for all compact subset.\n\n$$lim inf n→∞ 1 a n log P n (∃ν ∈ N n n : V n,an ν ∈ G) -inf g∈G S f (g).$$\n\n## Proof of Theorem 9.\n\nThere is no more different that need to be dealt with specifically. Following the Section 7 in [20], we can prove it directly. Now we are going to prove Theorem 3.\n\nProof of Theorem 3. Due to the many-to-one formula, for any compact subset K ⊂ C 0 ([0, 1]), we have\n\nThen, due to Lemma 2, Assumption 2 (f (1) < ∞) and the goodness of I 1 (•), we can obtain the the exponential tight property about the (sub-additive) measure {P n (∃ν ∈ N n n : V n,an ν ∈ •)} n 1. Using the Theorem 9, together with the exponential tight property, following the proof in Lemma 1.2.18 in Chapter 1 of [14], we can prove Theorem 3.\n\n$$P n (∃ν ∈ N n n : V n,an ν ∈ K) E n (# {ν ∈ N n n : V n,an ν ∈ K}) = exp(S n n )P n (B an n (•) ∈ K) Here B an n (•) is defined in Lemma 2. Then lim sup n→∞ 1 a n log P n (∃ν ∈ N n n : V n,an ν ∈ K) f (1) + lim sup n→∞ 1 a n log P n (B an n (•) ∈ K)$$\n\n## 2.6 The maximal displacement\n\nIn this subsection, we want to estimate the maximal displacement M n n of the time-inhomogeneous branching random walk under P n, and prove a limit theorem.\n\nFor the continuous function f in Assumption 2, define\n\nFor the n-th branching random walk, the maximal displacement in time n, i.e. M n n := max{V ν ; ν ∈ N n n }:\n\nTheorem 10. Under Assumption 2 and Assumption 3, the maximal displacement of the timeinhomogeneous branching random walk satisfies\n\nRemark 8. For normal distribution N (0, 1), the condition lim n→∞ an n = 0 about {a n } in Assumption 2 can be omitted (see Remark 2). In [19], Hammersley proved the almost surely limit for the maximal displacement at time n of supercritical branching random walk by using the sub-additive ergodic theorem. If the jump distribution is N (0, 1), and take a n = n, then Theorem 10 is a kind of weak law of large number for supercritical branching random walk. In detail, assume the mean of offspring distribution is m > 1, it is easy to calculate that the log-Laplacian transform of the offspring point process is Λ(λ) = log m + 1 2 λ 2, accounting to the law of large number in [19], then\n\nIn the setting of Theorem 10, take a n = n and f (t) = t log m, it is not hard to verify the supercritical branching random walk does satisfy Assumption 2 and Assumption 3, then solve eq. ( 23), we know A f = √ 2 log m, then according to Theorem 10, M n n =⇒ √ 2 log m, which also means the convergence in probability.\n\nProof of Theorem 10. For any ǫ > 0, let C := {g ∈ C 0 ([0, 1]); g(1) A f + ǫ}, and C is a closed subset of C 0 ([0, 1]). We claim that inf g∈C S f (g) > 0. Otherwise, if inf g∈C S f (g) = 0, there exist a sequence function\n\nDue to S f is a good rate function, then there is some sub-sequence function {g n k } that convergence to some g uniformly and S f (g) = 0. Because C is a closed set, g ∈ C, then g(1) A f + ǫ, which contradicts the definition of A f. According to the deviation principle, we have lim sup\n\nThen\n\nNext, for any ǫ > 0, we want to prove\n\nAccording to Assumption 2, there exist some h 0 ∈ (0, 1), such that f (h) < ǫ 2 4 for any h ∈ (0, h 0 ), then for any h ∈ (0, h 0 ),\n\nHere B an n (•) is defined in Lemma 2. Under Assumption 2, due to the choice of h, which means for n large enough, we have S n ⌊nh⌋ < anǫ 2 3, and for normal distribution, we know P(N (0, δ 2 ) < -x) < exp(-x 2 2δ 2 ) for any x > 0, as n → ∞, we have\n\nSo, for any ǫ > 0, there exist h 0 ∈ (0, 1), such that for any h ∈ (0, h 0 ), we have\n\nFor each n, according to the definition of A f, there exist a function\n\nn. According to the goodness of S f, there exist some sub-sequence {g n k } convergence to some function g in C 0 ([0, 1]), then g(1) = lim k→∞ g n k (1) = A f, and S f (g) = 0.\n\nFor the function g, define an increasing function Q(y) as follow:\n\nFor h small enough (in fact h < Q(1)), we can define φ(h) as the generalized inverse function of Q(y) as follow:\n\nAbout the function Q(y) and φ(h), we have Lemma 4. For any y > 0, Q(y) > 0, then φ(0) = 0.\n\nThe Lemma 4 will be proved after finishing the proof of Theorem 10. For h small enough such that f (h) < Q(1), define g h (s) ∈ C 0 ([0, 1]) with deviation as follow:\n\nThus g h (1) = g(1)g(φ(f (h))). Due to S f (g) = 0, according to the definition of φ, then φ(f (h)) h, and\n\nAnd for any γ ∈ N n ⌊nh⌋, for any ǫ > 0, define\n\nHere γ ≺ ν means γ is an ancestor of ν. According to branching property, we know above representation doesn't depend on the choose of γ, so just denote the value as ψ n (h, g, ǫ).\n\nIt is not hard to verified the deviation principle also works for the time-inhomogeneous branching random walk that roots at any γ ∈ N n ⌊nh⌋ (just repeat the previous argument). So, according to the deviation principle, for any open set\n\nThen, we know\n\nWhat's more,\n\nNote that, for any β > 0,\n\nUnder Assumption 3, we know\n\nUnder Assumption 2, and eq. ( 27), we know for any β > 0,\n\nCombine eqs. ( 28) to (31), firstly let n → ∞, then β → 0, we have\n\nAccording to Lemma 4 and the continuity of f, then lim h→0 + φ(f (h)) = φ(0 + ) = 0, so there exist some h 1 > 0 such that φ(f (h)) < 1 2 ǫ for any h ∈ (0, h 1 ). According to Assumption 2, there exist some h 2 > 0 such that f (h) < ǫ for any h ∈ (0, h 2 ). Together with the h 0 of eq. ( 26), fix any h ∈ (0, h 0 ∧ h 1 ∧ h 2 ), then g(φ(f (h))) < 2ǫ, and recall that g h (1) = g(1)g(φ(f (h))), thus\n\nCombine eqs. ( 26), ( 32) and ( 33), let n → ∞, we have\n\nDue to the arbitrariness of ǫ, together with eq. ( 12), then\n\nThe proof of Theorem 10 is finished. Now, we are going to prove Lemma 4:\n\nProof of Lemma 4. It is easy to find that g ′ (s) 0 almost surely, which means g is nondecreasing function (else, just set g ′ 1 = |g ′ |, then the function g 1 also satisfies the condition in eq. ( 23) while g 1 (1) > g(1) = A f, which contradicts the definition of A f ).\n\nDefine t 0 := sup{t ∈ [0, 1]; Q(t) = 0}, we will prove that t 0 = 0. If not, assume that t 0 > 0, then g(t) = 0 for all t ∈ [0, t 0 ] and\n\nis a continuous and strict positive function, and lim\n\n∞ (f (t) > 0 for all t > 0). So there exist some γ > 0 such that R(t) > γ for all t ∈ ( t 0 2, t 0 ]. Note that, f (t) -Q(t) is continuous function, and Q(t 0 ) = 0 and f (t 0 ) > 0, then we can fix some t 1 ∈ (t 0, 1), such that\n\nFix some δ ∈ (0, 2f (t 0 )\n\n). According to the choose of δ, the set\n\nis non-empty, and then fix some α ∈ δ 2 t 0 4Q(t 1 ), δt 0 2g(t 1 ) ⊂ (0, 1). Due to the choice of δ and α,\n\nConstruct a function g 1 with derivative g ′ 1 as follow:\n\nJust calculate:\n\nThus S f (g 1 ) = 0 and\n\nwhich contradicts the definition of A f. Thus t 0 = 0. Then for any y > 0, Q(y) > 0. And for the inverse function φ(h), due to the continuity of Q(y), it is not hard to prove lim\n\nThe proof of Lemma 4 is over.\n\n$$A f := sup g(1) : ˆr 0 1 2 g ′ (s) 2 ds f (r), ∀r ∈ [0, 1](23)$$\n\n$$M n n √ na n =⇒ A f, as n → ∞.$$\n\n$$M n n a.s -→ inf Λ(λ) λ = √ 2 log m.$$\n\n$${g n } ⊂ C such that lim n→∞ S f (g n ) = 0.$$\n\n$$n→∞ 1 a n log P n M n n √ na n A f + ǫ = lim sup n→∞ 1 a n log P n (∃ν ∈ N n n ; V n,an ν ∈ C) -inf g∈C S f (g) < 0.$$\n\n$$lim n→∞ P n M n n √ na n > A f + ǫ = 0. (24$$\n\n$$)$$\n\n$$lim n→∞ P n M n n √ na n A f -4ǫ = 1. (25$$\n\n$$)$$\n\n$$P n (∃ν ∈ N n n, V n,an ν (h) -ǫ) P n ∃ν ∈ N n ⌊nh⌋, V n ν √ na n -ǫ E n # ν ∈ N n ⌊nh⌋, V n ν √ na n -ǫ = exp S n ⌊nh⌋ P n (B an n (h) -ǫ)$$\n\n$$lim n→∞ P n (∀ν ∈ N n n, V n,an ν (h) -ǫ) lim n→∞ exp S n ⌊nh⌋ P n (B an n (h) -ǫ) lim n→∞ exp a n ǫ 2 3 exp - ǫ 2 a n 2h lim n→∞ exp a n ǫ 2 3 exp - a n ǫ 2 2 = 0.$$\n\n$$lim n→∞ P n (∀ν ∈ N n n, V n,an ν (h) > -ǫ) = 1. (26$$\n\n$$)$$\n\n$$g n ∈ C 0 [0, 1] such that S f (g n ) = 0 and A f g n (1) > A f -1$$\n\n$$Q(y) := ˆy 0 1 2 g ′ (s) 2 ds.$$\n\n$$φ(h) := inf {y ∈ [0, 1]; Q(y) > h}.$$\n\n$$g ′ h (t) := 0, t ∈ [0, φ(f (h))]; g ′ (t), t ∈ [φ(f (h)), 1].$$\n\n$$f (h) = Q(φ(f (h))).$$\n\n$$ψ n (h, g, ǫ)(γ) := P n (∃ν ∈ N n n, γ ≺ ν, V n,an ν (s) -V n,an ν (h) > g h (s) -ǫ, ∀s ∈ [h, 1] | F ⌊nh⌋$$\n\n$$V in C 0 ([0, 1 -h]; R), we know lim inf n→∞ 1 a n logP n (∃ν ∈ N n n, γ ≺ ν, V n,an ν (h + •) -V n,an ν (h) ∈ V ) -inf k∈V sup ˆω 0 1 2 k ′ (s) 2 ds -(f (h + ω) -f (h)) Choosing V := {k(•) ∈ C 0 ([0, 1 -h]; R) : k(s) > g h (h + s) -ǫ, ∀s ∈ [0, 1 -h]} and g h (h + •) ∈ V. Note that ˆr 0 1 2 g ′ h (h + s) 2 ds -f (h + r) + f (h)              0 -f (h + r) + f (h) 0, r ∈ [0, φ(f (h)) -h); ´r 0 1 2 g ′ (s) 2 ds -f (h + r) 0, r ∈ [φ(f (h)) -h, 1 -h].$$\n\n$$lim n→∞ 1 a n log ψ n (h, g, ǫ) = 0. (27$$\n\n$$)$$\n\n$$P n (∃ν ∈ N n n, V n,an ν (s) -V n,an ν (h) > g h (s) -ǫ, ∀s ∈ [h, 1]) =P n   γ∈N n ⌊nh⌋ {∃ν ∈ N n n, γ ≺ νV n,an ν (s) -V n,an ν (h) > g h (s) -ǫ, ∀s ∈ [h, 1]}   =E n 1 -(1 -ψ n (h, g, ǫ)) Z n ⌊nh⌋ 1 -E n exp -Z n ⌊nh⌋ ψ n (h, g, ǫ)(28)$$\n\n$$E n exp -Z n ⌊nh⌋ ψ n (h, g, ǫ) = E n exp -exp log Z n ⌊nh⌋ + log ψ n (h, g, ǫ) P n Z n ⌊nh⌋ exp βS n ⌊nh⌋ + E n exp -exp log Z n ⌊nh⌋ + log ψ n (h, g, ǫ) ; Z n ⌊nh⌋ > exp βS n ⌊nh⌋ P n Z n ⌊nh⌋ exp βS n ⌊nh⌋ + exp -exp βS n ⌊nh⌋ + log ψ n (h, g, ǫ)(29)$$\n\n$$lim β→0 lim sup n→∞ P n Z n ⌊nh⌋ exp βS n ⌊nh⌋ = 0 (30$$\n\n$$)$$\n\n$$lim n→∞ E n exp -exp βS n ⌊nh⌋ + log ψ n (h, g, ǫ) = 0. (31$$\n\n$$)$$\n\n$$lim n→∞ P n (∃ν ∈ N n n, V n,an ν (s) -V n,an ν (h) > g h (s) -ǫ, ∀s ∈ [h, 1]) 1 -lim β→0 lim sup n→∞ E n exp -Z n ⌊nh⌋ ψ n (h, g, ǫ) = 1. (32$$\n\n$$) About φ(f (h)), |g(φ(f (h)))| 2 = ˆφ(f(h)) 0 g ′ (s)ds 2 ˆφ(f(h)) 0 g ′ (s) 2 ds ˆφ(f(h)) 0 1ds 2f (h)φ(f (h)).$$\n\n$$P n M n n √ na n g(1) -4ǫ P n M n n √ na n g(1) -g(φ(f (h))) -2ǫ P n (∃ν ∈ N n n, V n,an ν (s) -V n,an ν (h) > g h (s) -ǫ, ∀s ∈ [h, 1]; ∀ν ∈ N n n, V n,an ν (h) > -ǫ) P n (∃ν ∈ N n n, V n,an ν (s) -V n,an ν (h) > g h (s) -ǫ, ∀s ∈ [h, 1]) + P n (∀ν ∈ N n n, V n,an ν (h) > -ǫ) -1(33)$$\n\n$$lim n→∞ P n M n n √ na n g(1) -4ǫ = 1.(34)$$\n\n$$M n n √ na n =⇒ A f.$$\n\n$$Q(t) > 0 for t ∈ (t 0, 1]. Let R(t) := f (t) t -1 2 t 0 in t ∈ ( t 0 2, t 0 ]$$\n\n$$t→ t 0 2 + R(t) =$$\n\n$$f (t) -Q(t); t ∈ [t 0, t 1 ] f (t 0 ) 2 for all t ∈ [t 0, t 1 ].$$\n\n$$t 0 ∧ √ 2γ ∧ 2Q(t 1 ) g(t 1 ) ∧ 2g(t 1 ) t 0$$\n\n$$δ 2 t 0 4Q(t 1 ), δt 0 2g(t 1 )$$\n\n$$1 2 δ 2 (t - t 0 2 ) < f (t), ∀t ∈ [ t 0 2, t 0 ]; αg(t 1 ) < δt 0 2 ; δ 2 t 0 4 < f (t 0 ) 2 < f (t) -Q(t), ∀t ∈ [t 0, t 1 ]; δ 2 t 0 4 < αQ(t 1 ).$$\n\n$$g ′ 1 (t) :=              0, t ∈ [0, t 0 2 ] δ, t ∈ ( t 0 2, t 0 ] (1 -α)g ′ (t), t ∈ (t 0, t 1 ] g ′ (t), t ∈ (t 1, 1].$$\n\n$$ˆt 0 1 2 g ′ 1 (s) 2 ds =                      0 f (t), t ∈ [0, t 0 2 ] 1 2 δ 2 t -1 2 t 0 f (t), t ∈ ( t 0 2, t 0 ] (1 -α) 2 Q(t) + δ 2 t 0 4 < (1 -α) 2 Q(t) + f (t) -Q(t) f (t), t ∈ (t 0, t 1 ] (1 -α) 2 Q(t 1 ) + δ 2 t 0 4 + Q(t) -Q(t 1 ) Q(t) + δ 2 t 0 4 -αQ(t 1 ) < Q(t) f (t), t ∈ (t 1, 1].$$\n\n$$g 1 (1) = ˆ1 0 g ′ 1 (s)ds = 0 + δt 0 2 + (1 -α)g(t 1 ) + g(1) -g(t 1 ) = g(1) + δt 0 2 -αg(t 1 ) > g(1) = A f.$$\n\n$$h→0 + φ(h) = t 0 = 0.$$\n\n## 3 Critical branching processes in random environment\n\nIn this section, we will consider the critical branching processes in random environment. In order to use the results that established in Section 2 to help us investigate the conditional limit theorem, we just need to verify that conditioned on the survival events, the reduced branching processes do satisfy the assumptions in Section 2.1.\n\n## For an environment\n\n. For any k n, let Z(k, n) be the number of particles at time k which have at least one descendant survived at time n. Given the environment ξ, and conditional on {Z n > 0}, the reduced processes is a time in-homogeneous branching process denoted by\n\n> 0}, and its branching mechanism is given by F r,n = {F r,n 1, F r,n 2, • • •, F r,n n } (here r means \"reduced\"), the generating function of F r,n k has following representation (in fact, we can calculated it by the compound binomial distribution with condition):\n\nAlso see [17,10]. Recall eqs. ( 6) and ( 7), take derivative in eq. ( 35), we have\n\nThe mean of the reduced processes is defined as\n\n, which depends only on the environment ξ and satisfies\n\nHere S k is defined in eq. ( 7).\n\nDefine L(k, n) := min{S j : k j n} be the minimal value of path {S k, 0 k n} between k and n. Firstly, recall some basic theorems describe the behavior of branching processes in random environment, which is related to the Brownian meander.\n\nTheorem 11 (Theorem 1.5 in [5]). Under Assumption 1, the following convergence holds in D([0, 1]):\n\nHere W + t ; t ∈ [0, 1] is the Brownian meander.\n\nTheorem 12 (Corollary 1.6 in [5]). Under Assumption 1, the following convergence holds in D([0, 1]):\n\nHere W + t ; t ∈ [0, 1] is the Brownian meander.\n\nRemark 9. Theorem 11 and Theorem 12 is established for linear fractional case in [2], [3] and many other articles for some wild cases. What's more, in [5], Afanasyev, Geiger, Kersting and Vatutin investigated a new method, and extend such results to more general condition, especially under Assumption 1.\n\nNow we are going to verified that for reduced branching processes in random environment, Assumption 2 in Section 2.1 is satisfied.\n\nLemma 5. Under Assumption 1, for any ǫ > 0,\n\nProof of Lemma 5. For any ǫ > 0, define\n\nT n := max\n\nHere κ k (a) is defined in Assumption 1. Set L n := min{S 0, S 1, • • •, S n }, note that\n\nin D([0, 1]), where W + t is the Brownian meander, thus its path is continuous almost surely. So about the events W n, lim\n\nAbout events T n, we will use the structure that established in [5] to estimate the probability. Note that P + denote the corresponding Doob-h transform of P in [5], it is called the random walk condition to stay non-negative forever (see Section 2 in [5] for more details). According to the proof of Lemma 2.7 in [5], there exist δ ′ > 0, such that\n\n$$ξ := {F 1, F 2, • • • }, we also use F k denote the corresponding generating function. Define F k,n (s) :=      F k+1 • F k+2 • • • • F n, k < n; δ 1 (s), k = n; F k • F k+1 • • • • F n+1, k > n. Then E ξ [Z n = 0|Z k = 1] = F k,n (0) for any k n, define P ξ (k, n) := 1 -F k,n(0)$$\n\n$${Z(0, n), Z(1, n), • • •, Z(n, n)|Z n$$\n\n$$F r,n k (s) : = E ξ [s Z(k,n) |Z k-1 = 1, Z n > 0] = 1 E ξ [Z n > 0|Z k-1 = 1] ∞ j=0 E ξ [s Z(k,n) ; Z k = j; Z(k, n) > 0|Z k-1 = 1]. = F k (1 -P ξ (k, n) + sP ξ (k, n)) -1 + P ξ (k -1, n) P ξ (k -1, n)(35)$$\n\n$$F r,n k := P ξ (k, n) P ξ (k -1, n) exp(X k ); F r,n k := P ξ (k -1, n)η k. (36$$\n\n$$)$$\n\n$$E ξ [Z(k, n)|Z n > 0]$$\n\n$$E ξ [Z(k, n)|Z n > 0] = k j=1 F r,n k = P ξ (k, n) P ξ (0, n) exp(S k ). (37$$\n\n$$)$$\n\n$$L S ⌊nt⌋ σ √ n ; t ∈ [0, 1] |Z n > 0 =⇒ L W + t ; t ∈ [0, 1], as n → ∞.$$\n\n$$L log Z ⌊nt⌋ σ √ n ; t ∈ [0, 1] |Z n > 0 =⇒ L W + t ; t ∈ [0, 1], as n → ∞.$$\n\n$$lim n→∞ P log 1 + n k=1 η k > ǫσ √ n | Z n > 0 = 0.$$\n\n$$W n := max 1 k n |X k | > ǫ √ n4$$\n\n$$1 k n log + κ k (a) > ǫ √ n 4.$$\n\n$$S ⌊nt⌋ σ √ n ; t ∈ [0, 1]|L n 0 =⇒ W + t ; t ∈ [0, 1]$$\n\n$$n→∞ P (W n |L n 0) = 0. (38$$\n\n$$)$$\n\n$$κ k (a) = O exp k 1 2 -δ ′, P + a.s.$$\n\n## Thus we know\n\n½ {T n } → 0, P + a.s.\n\nAnd due to dominating convergence theorem, above convergence also holds in mean. Due to Lemma 2.5 in [5],\n\nFor any k 1, we have η k κ k (a) + a exp(-X k ).\n\nNote that, for fixed constant Γ, for n large enough, such that exp ǫσ\n\nHence, together with eqs. ( 38) and (39), we know for any constant Γ,\n\nDefine τ n := min {0 k n; S k = L n }, and L k,n := min\n\nto the dominating convergence theorem, and lim\n\nAccording to the dominating convergence theorem, combine eqs. ( 8) and (40), thus\n\nThe proof of Lemma 5 is over.\n\nAbout the reduced branching processes in random environment, we have Theorem 13 (Theorem 3 in [33]). Under Assumption 1, condition on {Z n > 0}, in D([0, 1]), about the reduced branching processes in random environment {Z(k, n)},\n\nHere Λ t := inf{W + s ; s ∈ [t, 1]}, where W + t ; t ∈ [0, 1] is the Brownian meander.\n\nRemark 10. In [33], Vatutin proved Theorem 13 under more stronger assumptions (comparing with Assumption 1). At that time, the technology in [5] was not yet available, so the results about the random walk in random environment actually need more stronger assumptions. Together with the methods established in [5], by repeat the steps in [33], we can expand related results to Assumption 1.\n\nIn this article, we need a more stronger results about the critical reduced branching processes in random environment. Theorem 14. Under the Assumption 1, following convergence holds in D([0, 1]; R 3 ):\n\nHere Λ t := inf{W + s ; s ∈ [t, 1]}, where W + t ; t ∈ [0, 1] is the Brownian meander.\n\nRemark 11. Condition on the survival events {Z n > 0}, the reduced branching processes\n\n, can be generated by following steps:\n\n• Firstly, use P ξ (0, n) to bias the environment ξ, that is for any A ⊂ P(N 0 ) N is a measurable set, define Q n (A) = P(P ξ (0, n); A) P(Z n > 0), then Q n is a probability on P(N 0 ) N. According to law Q n, generate an environment ξ.\n\n• Given the environment ξ, get the reduced branching mechanism F r,n (ξ\n\n)).\n\n• Then, drive a branching process according to the reduced branching mechanism F r,n (ξ) until time n, and the process has the same law as the reduced branching processes.\n\nProof of Theorem 14. For ǫ > 0, define\n\nProof of lemma 6. Note that for any k l n,\n\nHence, we have\n\nand due to Agresti?s estimate (see the Section 2 in [18], or eq (3.4) in [5]), we have\n\nCombine eqs. ( 41) and (42) and S 0 = 0,\n\nNote that L(0, n) S 0 = 0, and log(1\n\nHence,\n\nAccording to Theorem 11 and Lemma 5, then Lemma 6 is proved. Lemma 7. For any ǫ > 0, the event R n (ǫ), satisfies lim\n\nFirstly, we will prove that for any fixed t ∈ [0, 1],\n\nDefine:\n\nDue to (41), on the event\n\nThen,\n\nHere\n\nNote that, condition on F ⌊nt⌋ (here F k denote as the σ algebra generated by total environment and the branching processes up to time k), Z(⌊nt⌋, t) can be view as the binomial distribution with parameter Z ⌊nt⌋, P ξ (⌊nt⌋, n). So,\n\nAnd due to Equation ( 8 (44)\n\nAccording to Theorem 11 and Theorem 13, for any x > 0,\n\nHere Λ t := inf{W + s ; s ∈ [t, 1]}, where W + t ; t ∈ [0, 1] is a Brownian meander. Combine eqs. ( 44) and (45), we will prove lim n→∞ P (χ n (t) < -ǫ|Z n > 0) = 0. Firstly, for any δ > 0, there exist γ > 0 and K, such that for n large enough,\n\nNote that\n\nFor any y > ǫ, and ǫ 1 > 0,\n\nLet n k, be any increasing sequence of integers such that the limit The proof of Lemma 7 is over.\n\nAccording to Lemma 5, Lemma 6 and Lemma 7, together with Theorem 13, then Theorem 14 is proved.\n\n$$lim n→∞ P (T n |L n 0) = 0. (39$$\n\n$$)$$\n\n$$√ n 2 -Γ an n, then B n (Γ) := log Γ + n k=1 η k > ǫσ √ n ⊂ n max 1 k n κ k + an max 1 k n exp(-X k ) > exp ǫσ √ n -1 ⊂ max 1 k n κ k + max 1 k n exp(-X k ) > exp ǫσ √ n 2 ⊂ max 1 k n log + κ k > ǫσ √ n 4 ∪ max 1 k n |X k | > ǫσ √ n4$$\n\n$$lim n→∞ P (B n (Γ)|L n 0) = 0.$$\n\n$$0 i n-k {S k+i -S k }. Note that P (B n (1); Z n > 0) = n k=0 P (B n (1); Z n > 0; τ k = k; L k,n 0) ∞ k=0 P (B n (1); Z n > 0; τ k = k; L k,n 0) ½ {k n} (40) For k ∈ N fixed, define Γ k := k+ k i=1 η i < ∞, it is a F k measurable random variable, according$$\n\n$$n→∞ √ nP(L n 0) = K 1 ∈ (0, ∞), then lim n→∞ √ nP (B n (1); Z n > 0; τ k = k; L k,n 0) lim n→∞ √ nE P log Γ k + n i=k+1 η i > ǫσ √ n|L k,n 0 ; τ k = k P(L n,k 0) K 1 lim n→∞ E (P (B n-k (Γ k )|L n-k 0)) K 1 E lim n→∞ P (B n-k (Γ k )|L n-k 0) = 0.$$\n\n$$lim n→∞ P(B n (1)|Z n > 0) = 0.$$\n\n$$L Z(⌊nt⌋, n) σ √ n ; t ∈ [0, 1] |Z n > 0 =⇒ L ({Λ t ; t ∈ [0, 1]}), as n → ∞.$$\n\n$$L log E ξ [Z(⌊nt⌋, n) | Z n > 0] σ √ n, log Z(⌊nt⌋, n) σ √ n, L(nt, n) σ √ n ; t ∈ [0, 1] | Z n > 0 =⇒ L ({Λ t, Λ t, Λ t ; t ∈ [0, 1]}) as n → ∞.$$\n\n$${Z(0, n), Z(1, n), • • •, Z ( n, n)|Z n > 0}$$\n\n$$) := {F r.n 1 (ξ), • • •, F r,n k (ξ), • • •, F r,n n (ξ)} (see eq. (35$$\n\n$$H n (ǫ) := ∃t ∈ [0, 1], log E ξ [Z(⌊nt⌋, n) | Z n > 0] -L(⌊nt⌋, n) σ √ n > ǫ R n (ǫ) := ∃t ∈ [0, 1], log Z (⌊nt⌋, n) -L(⌊nt⌋, n) σ √ n > ǫ Lemma 6. For any ǫ > 0, the event H n (ǫ), satisfies lim n→∞ P (H n (ǫ) | Z n > 0) = 0$$\n\n$$P ξ (k, n) P ξ (k, l) E ξ (Z l |Z k = 1) = exp(S l -S k ).$$\n\n$$log P ξ (k, n) min{S l -S k ; k l n} = L(k, n) -S k.(41)$$\n\n$$log P ξ (k, n) L(k, n) -S k -log   1 + n j=k+1 η j  (42)$$\n\n$$-L(0, n) -log   1 + n j=1 η j  . -log P ξ (0, n) -log   1 + n j=k+1 η j   -log P ξ (0, n) + log P ξ (k, n) + S k -L(k, n) = log E ξ [Z(⌊nt⌋, n) | Z n > 0] -L(k, n) -log P ξ (0, n) -L(0, n) + log   1 + n j=1 η j  .$$\n\n$$+ n j=1 η j ) 0, therefore log E ξ [Z (⌊nt⌋, n) | Z n > 0] σ √ n - L(⌊nt⌋, n) σ √ n |L(0, n)| + log 1 + n k=1 η k σ √ n$$\n\n$$H n (ǫ) ⊂        |L(0, n)| + log 1 + n k=1 η k σ √ n > ǫ        ⊂ |L(0, n)| ǫσ √ n 2 ∪ log 1 + n k=1 η k > ǫσ √ n2$$\n\n$$n→∞ P (R n (ǫ) | Z n > 0) = 0. Proof of Lemma 7. Denote χ n (t) := log Z(⌊nt⌋, n) -L(⌊nt⌋, n) σ √ n$$\n\n$$lim n→∞ P (|χ n (t)| > ǫ|Z n > 0) = 0.(43)$$\n\n$$A n := exp S ⌊nt⌋ - ǫ 2 σ √ n Z ⌊nt⌋ exp S ⌊nt⌋ + ǫ 2 σ √ n$$\n\n$$A n ∩ {χ n (t) > ǫ}, Z (⌊nt⌋, n) > exp L(⌊nt⌋, n) + ǫσ √ n Z ⌊nt⌋ P ξ (⌊nt⌋, n) exp ǫσ √ n2$$\n\n$$P (½ {χ n > ǫ} ½ {A n } |Z n > 0) P (A n ∩ C n |Z n > 0)$$\n\n$$C n := {Z (⌊nt⌋, n) > Z ⌊nt⌋ P ξ (⌊nt⌋, n) exp ǫσ √ n 2 }.$$\n\n$$P (½ {A n } ½ {C n } ½ {Z n > 0}) = P P ½ {A n } ½ {C n } ½ {Z n > 0} |F ⌊nt⌋ = P ½ {A n } P ½ {C n } |F ⌊nt⌋ P ½ {A n } exp - ǫσ √ n 2 exp - ǫσ √ n2$$\n\n$$lim n→∞ P log Z (⌊nt⌋, n) σ √ n > x|Z n > 0 = lim n→∞ P χ n (t) + L(⌊nt⌋, n) σ √ n > x|Z n > 0 = lim n→∞ P L(⌊nt⌋, n) σ √ n > x|Z n > 0 = P (Λ t > x).(45)$$\n\n$$P L(⌊nt⌋, n) σ √ n < γ|Z n > 0 + P L(⌊nt⌋, n) σ √ n > γ + Kǫ|Z n > 0 < δ.(46)$$\n\n$$P (χ n (t) < -ǫ|Z n > 0) = P χ n (t) < -ǫ, L(⌊nt⌋, n) σ √ n < γ|Z n > 0 + P χ n (t) < -ǫ, L(⌊nt⌋, n) σ √ n γ + Kǫ|Z n > 0 + K k=1 P χ n (t) < -ǫ, L(⌊nt⌋, n) σ √ n ∈ [γ + (k -1)ǫ, γ + kǫ) |Z n > 0(47)$$\n\n$$P χ n (t) + L(⌊nt⌋, n) σ √ n > y|Z n > 0 = P χ n (t) + L(⌊nt⌋, n) σ √ n > y; χ n (t) < -ǫ|Z n > 0 + P χ n (t) + L(⌊nt⌋, n) σ √ n > y; χ n (t) ∈ [-ǫ,$$\n\n## 4 Proof of Theorems\n\nIn this section, we will prove the Theorem 1 and Theorem 2.\n\n## 4.1 Proof of Theorem 1\n\nIn this subsection, We will use the coupling methods to prove Theorem 1. Due to eq. (36), Lemma 5, Theorem 14 and Remark 11, use the Skorohod representation theorem, we can construct a probability space (Ω, F, Q), such that Here W + 0 (t) is a normal Brownian excursion and α is a independent random variable uniformly distribution on (0, 1).\n\nRemark 12. Compare with the Theorem 5 in [4], the assumption in Theorem 15 is much weaker. At that time, the assumptions in [4] is just to ensure the Theorem 12 was correct. As the same argument in Remark 10, we can repeat the proof in [4] to extend related results for Assumption 1.\n\nProof of Lemma 9. For any ǫ > 0, under Assumption 1, by the moment estimate, we have Together with eqs. ( 41) and (42), the proof is over.\n\nFor any x ∈ (0, 1) and y > 0, the inequality (1x) y exp(-xy) holds, then\n\nTake exception in both side, then\n\nLet n → ∞, then lim sup\n\nNext take ǫ → 0, and note that lim What's more, use eq. ( 65) and repeat the proof in [4] (the Lemma 4 is replaced by eq. ( 65), and other arguments are consistent), we can prove that there exist some finite positive constant K 2, such that Hence, the right hand of eq. ( 10) is proved. Now, the proof of Theorem 2 is finished.\n\n$$P ξ (Z nǫ = 0) exp(σ √ n) (1 -P ξ (Z nǫ > 0)) exp(σ √ n) ½ P ξ (Z nǫ > 0) exp - σ √ n 2 + ½ P ξ (Z nǫ > 0) < exp - σ √ n 2. exp -exp σ √ n 2 ½ P ξ (Z nǫ > 0) exp - σ √ n 2 + ½ P ξ (Z nǫ > 0) < exp - σ √ n 2$$\n\n$$E P ξ (Z nǫ = 0) exp(σ √ n) exp -exp σ √ n 2 P 1 σ √ n log (1 -F 0,nǫ (0)) - 1 2 + P 1 σ √ n log (1 -F 0,nǫ (0)) < - 1 2$$\n\n$$n→∞ E P ξ (Z nǫ = 0) exp(σ √ n) P inf t∈[0,ǫ] B(t) < -1 2.$$\n\n$$lim n→∞ √ nP sup k 1 Z n > exp σ √ n = K 2. (66$$\n\n## References\n\n1. Addario, Berry, Reed (2009) \"Minima in branching random walks\" *Ann. Probab*\n\n2. Afanasyev (1993) \"A limit theorem for a critical branching process in a random environment\" *Diskret. Mat*\n\n3. Afanasyev (1997) \"A new limit theorem for a critical branching process in a random environment\" *Diskret. Mat*\n\n4. Afanasyev (1999) \"On the maximum of a critical branching process in a random environment\" *Diskret. Mat*\n\n5. Afanasyev, Geiger, Kersting et al. (2005) \"Criticality for branching processes in random environment\" *Ann. Probab*\n\n6. Aïdékon (2013) \"Convergence in law of the minimum of a branching random walk\" *Ann. Probab*\n\n7. Bachmann (2000) \"Limit theorems for the minimal position in a branching random walk with independent logconcave displacements\" *Adv. in Appl. Probab*\n\n8. Biggins (1976) \"The first-and last-birth problems for a multitype age-dependent branching process\" *Advances in Appl. Probability*\n\n9. Biggins, Kyprianou (2004) \"Measure change in multitype branching\" *Adv. in Appl. Probab*\n\n10. Borovkov, Vatutin (1997) \"Reduced critical branching processes in random environment\" *Stochastic Process. Appl*\n\n11. Bramson, Ding, Zeitouni (2016) \"Convergence in law of the maximum of nonlattice branching random walk\" *Ann. Inst. Henri Poincaré Probab. Stat*\n\n12. Bramson, Zeitouni (2009) \"Tightness for a family of recursion equations\" *Ann. Probab*\n\n13. Bramson (1978) \"Minimal displacement of branching random walk\" *Z. Wahrsch. Verw. Gebiete*\n\n14. Dembo, Zeitouni (1998) \"Large deviations techniques and applications\"\n\n15. Fang, Zeitouni (2012) \"Branching random walks in time inhomogeneous environments\" *Electron. J. Probab*\n\n16. Fang, Zeitouni (2012) \"Slowdown for time inhomogeneous branching Brownian motion\" *J. Stat. Phys*\n\n17. Fleischmann, Siegmund-Schultze (1977) \"The structure of reduced critical Galton-Watson processes\" *Math. Nachr*\n\n18. Geiger, Kersting (2000) \"The survival probability of a critical branching process in random environment\" *Teor. Veroyatnost. i Primenen*\n\n19. Hammersley (1974) \"Postulates for subadditive processes\" *Ann. Probability*\n\n20. Hardy, Harris (2006) \"A conceptual approach to a path result for branching Brownian motion\"\n\n21. Hu, Shi (2009) \"Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees\" *Ann. Probab*\n\n22. Huang, Liu (2014) \"Branching random walk with a random environment in time\"\n\n23. Kesten (1995) \"Branching random walk with a critical branching part\" *J. Theoret. Probab*\n\n24. Kingman (1975) \"The first birth problem for an age-dependent branching process\" *Ann. Probability*\n\n25. Kozlov (1976) \"The asymptotic behavior of the probability of non-extinction of critical branching processes in a random environment\" *Teor. Verojatnost. i Primenen*\n\n26. Lalley, Sellke (1987) \"A conditional limit theorem for the frontier of a branching Brownian motion\" *Ann. Probab*\n\n27. Lalley, Shao (2015) \"On the maximal displacement of critical branching random walk\" *Probab. Theory Related Fields*\n\n28. Maillard, Zeitouni (2016) \"Slowdown in branching Brownian motion with inhomogeneous variance\" *Ann. Inst. Henri Poincaré Probab. Stat*\n\n29. Bastien Mallein (2015) \"Maximal displacement in a branching random walk through interfaces\" *Electron. J. Probab*\n\n30. Bastien Mallein (2015) \"Maximal displacement of a branching random walk in time-inhomogeneous environment\" *Stochastic Process. Appl*\n\n31. Mallein, Mi L Oś (2019) \"Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment\" *Stochastic Process. Appl*\n\n32. Sawyer, Fleischman (1979) \"Maximum geographic range of a mutant allele considered as a subtype of a brownian branching random field\" *PNAS*\n\n33. Vatutin (2002) \"Reduced branching processes in a random environment: the critical case\" *Teor. Veroyatnost. i Primenen*<|endoftext|>"
+    },
+    "test": {
+      "total_tokens": 128912372,
+      "example": "# Entropy analysis and entropy stable DG methods for the shallow water moment equations\n\nJulio Careaga, Patrick Ersing, Julian Koellermeier, Andrew Winters\n\n## Abstract\n\nWe demonstrate that the shallow water moment equations satisfy an auxiliary entropy conservation law, where the entropy function corresponds to the total energy. Additionally, we show that the classical Newtonian slip friction and Manning friction terms are entropy dissipative with respect to the developed entropy variables. The results from the continuous entropy analysis are used to construct an entropy stable and well-balanced nodal discontinuous Galerkin spectral element method for the spatial approximation. Key to ensure the entropy stability of the scheme is the derivation of entropy conservative numerical fluxes that satisfy a discrete version of the entropy flux compatibility condition. Finally, numerical examples demonstrate the performance of the scheme and validate the theoretical results.\n\n## 1. Introduction\n\nShallow water equations (SWE) describe the evolution of a fluid under the assumption that the water thickness is orders of magnitude smaller than the horizontal scale. From the conservation of mass and momentum balance of the fluid for a free-boundary problem, the one-dimensional SWE consist of a two-by-two system for the water thickness and depth-averaged velocity. Despite the fact that SWE make a significant reduction in the difficulty of solving the incompressible Navier-Stokes equations, the reduction of the velocity profile to one depth-averaged variable may be overly strong and therefore lead to inaccuracies [1,2]. Shallow water moment equations (SWME) arise as an extension of the SWE, and assume that the velocity profile is polynomial in the vertical coordinate [3]. In SWME, the moments correspond to the coefficients of the polynomial expansion, which become part of the unknowns of the system together with the water height and the depth-averaged velocity. Therefore, increasing the order of this polynomial expansion enhances the accuracy when modelling velocity profiles with highly nonlinear variations with respect to depth. For a zero-order polynomial expansion the SWME coincides with the standard SWE. Further variants of the SWME have been proposed in the literature, two of the most relevant ones are: the hyperbolic shallow water moment equations in [4], and the shallow water linearized moment equations (SWLME) in [5]. These models are developed from the general SWME. The first discards some terms, in order to guarantee hyperbolicity and in the case of the second model, also the analytical computation of steady states. Other works have extended the SWME to non-hydrostatic flows [6], and to two-dimensional horizontal domains [7], while a general framework that extends the SWME to granular flows over inclined planes is proposed in [8].\n\nIn the standard SWE, the governing equations ensure conservation of mass, while the momentum balance is described by a depth-averaged velocity equation. To derive an energy equation, in [9], the authors use a skewsymmetric formulation of the SWE to obtain both the kinetic and potential energy equations, and the total energy as a result of their sum. Analogous to the SWE, the derived total energy corresponds to a convex mathematical entropy function for the SWME, which enables the construction of entropy stable schemes. In conservation laws, the mathematical entropy is related to the existence of physically relevant solutions, and also to the nonlinear stability of numerical schemes [10,11,12]. A methodology to derive entropy stable discontinuous Galerkin methods can be found in [13] and [14], where the authors use summation-by-parts properties and a flux-differencing formulation to derive entropy conservative fluxes. The numerical schemes developed in [15,16], use this approach for multilayer shallow water models, including nonconservative terms and ensuring the well-balancing property. In the same line, an entropy stable discontinuous Galerkin method in fluctuation form applied for the Saint-Venant-Exner system, is developed in [17]. In the context of SWLME, in [18], the authors develop a path-conservative and well-balanced discontinuous Galerkin scheme by introducing equilibrium variables.\n\nAs with SWE, the SWME satisfy the conservation of mass, while the momentum balance can be described from the equations for the depth-average velocity and moments. However, due to the nonlinear coupling between the mean velocity equation and moment equations, the derivation of a kinetic energy equation becomes non-trivial. Building on the procedure presented in [9] for the SWE, we derive an energy equation for the SWME.\n\nThe first aim of this work is to derive this energy equation for the SWME in the general form introduced in [3], where the total kinetic energy accounts for contributions from the moments. Unlike previous contributions made for the SWLME [19,18,20], we deal here with the nonlinear terms arising from the coupling between the average velocity and moments. Additionally, we demonstrate that the inclusion of two common friction terms has a dissipative contribution to the energy equation. The second aim is to develop an entropy stable discontinuous Galerkin numerical scheme that can handle the non-conservative products, a common feature of shallow water models, as well as step gradients due to possibly discontinuous solutions.\n\nThis paper is organized as follows. In Section 2, the kinetic, potential and total energy equations are derived for the SWME. In addition, a convex entropy function, entropy flux, and entropy variables are established as a consequence of the total energy formulation. In Section 2.1, we show that the Newtonian slip and Newtonian Manning friction terms are, in fact, entropy dissipative. The numerical approximation is described in Section 3. The discontinuous Galerkin scheme and corresponding conditions for semi-discrete entropy preservation are discussed in Section 3.1 and the construction of the entropy conservative numerical flux is treated in Section 3.2. In Section 3.3 we then construct an entropy stable scheme by adding suitable numerical dissipation at element interfaces. A series of illustrative numerical examples showing the performance and reliability of the developed numerical scheme are presented in Section 4. Conclusions and final remark are included in Section 5.\n\n## 2. Continuous entropy analysis for the SWME\n\nWe consider the one-dimensional shallow water moment equations (SWME), originally introduced in [3], for the case of frictionless conditions and including a variable bottom bathymetry. Given the spatial coordinate x ∈ Ω := (x a, x b ) with x a < x b, we assume that the bathymetry function is defined by the time-independent function b = b(x). The sought unknowns in the shallow water moment model are the water height h = h(x, t), the depth-averaged or mean velocity u m = u m (x, t), and the N moments α i = α i (x, t) for all x ∈ Ω and t > 0. The mean velocity together with the moments determine the horizontal component of the fluid velocity, denoted by u, through the following ansatz:\n\nwhere\n\nis the scaled and normalized vertical coordinate, z is the original vertical coordinate, and ϕ 1, ϕ 2,..., ϕ N are the first N shifted Legendre polynomials defined in the unit interval and normalized setting ϕ i (0) = 1 for all i = 1,..., N. The polynomials ϕ i are obtained from the classical Legendre polynomials\n\nThe first three shifted Legendre polynomials are\n\nIn turn, the SWME model is built from the assumption that the velocity of the fluid is polynomial on the vertical coordinate. The SWME in [3] describes the evolution of h, u m and α i for i = 1,..., N, and is formulated as follows:\n\nfor all x ∈ Ω and t > 0, where Ψ in (3b) is\n\nThe terms A i and B i in (3c) introduce additional nonlinearities to the system as they involve products of the moments and their partial derivatives, as well as factors of h. They are given by\n\nwhere the following coefficients are used\n\nNote that B i in (3c) corresponds to a non-conservative term. Defining the N-dimensional vectors α := (α 1, α 2,..., α N ) T, A := (A 1, A 2,..., A N ) T and B := (B 1, B 2,..., B N ) T, we can conveniently write (3c) as\n\nA variant of the SWME, the so-called shallow water linearized moment equations (SWLME), is developed in [5]. Therein, the model takes A = B = 0 so that\n\nIn what follows, we derive energy equations associated with (3). This is based on the hierarchical structure of the SWME: in the zeroth-order case, where all moments vanish, the system reduces to the SWE. Consequently, the energy equation derivation developed for the SWE in [9] can be directly applied to the system (3). The linearized model SWLME system (6) is hyperbolic and its eigenvalues are given by [5, Theorem 1]:\n\nUnlike the SWLME, the general SWME system (3) is only conditionally hyperbolic, and for certain values of the unknowns the hyperbolicity is lost [4].\n\nIn the next lemma, we provide an averaged kinetic energy equation for the mean velocity and a potential energy equation for system (3). Lemma 1. Let h, u m and α be a solution to (3). Then, the following averaged kinetic and potential energy equations hold\n\nwhere 1 2 hu 2 m is the kinetic energy in SWE, and 1 2 gh 2 + ghb is the potential energy. 3\n\nProof. To derive (8), we follow a similar approach as in the derivation of the kinetic energy equation in [9]. Subtracting (3a) multiplied by u m from the momentum equation (3b), we get\n\nNext, the arithmetic average between (3b) and (10) gives\n\nand multiplying this equation by u m and further grouping terms we arrive at the desired averaged kinetic energy equation (8). As (3a) coincides with the equation for conservation of mass in the SWE, the potential energy equation remains the same. Therefore, we omit its derivation and refer the reader to [9]. Now, before deriving an equation for the total kinetic energy, including the contributions by the N moment coefficients, we must first analyze the term containing ∂ x Ψ in the averaged kinetic energy equation (8). In the following auxiliary lemma, we provide a useful equation for the term Ψ from (4).\n\nLemma 2. The auxiliary variable Ψ defined by (4), which depends on the moments α, satisfies the following equation\n\nProof. Subtracting (3a) multiplied with α i from the moment equation (3c), we obtain\n\nNext, the result from (13) multiplied with α i plus (3a) multiplied by 1 2 α 2 i yields the following auxiliary equations for each i = 1,..., N\n\nwhere we have used the product rule and the identity\n\nThen, dividing (14) by 2i + 1 and summing over the moments from 1 to N, we obtain (12).\n\nEquation (12) from Lemma 2 can be seen as a partial kinetic energy equation. The total kinetic energy equation for the SWME can now be obtained by combining (8) and (12). However, the last term on the left-hand side of (12), which involves A i and B i, is not formulated in conservative form. Denoting this term by Q, we have\n\nwhere the unscaled coefficients are A i jk A i jk /(2i + 1) and B i jk B i jk /(2i + 1). The following lemma shows that Q can be written in conservative form, namely as the (spatial) partial derivative of a function. Lemma 3. The term Q can be written in conservative form as follows\n\nwhere\n\nProof. Computing the partial derivative of Q with respect to x in (17) and applying product rule, we have\n\nWe observe that the right-hand side of the above equation is equal to Q (15) provided the following identity holds\n\nAs shown in Appendix Appendix A, the condition (18) is true for all i, j, k = 1,..., N, which concludes the proof.\n\nThen, we define the vector of conservative variables as u = (h, hu m, hα 1, hα 2,..., hα N ) T ∈ R N+2 for the SWME (3) (or SWLME (6)), which in short notation can be written as u = h, hu m, hα T T. We are now in position to establish an equation for the total energy in the SWME model, and further determine the associated flux related to the energy balance.\n\nTheorem 1. Let h, u m and α be the components of the solution to system (3). Then, the total energy E is a conserved quantity with total energy equation\n\nwhere the total energy E and total flux F are defined as\n\nProof. We first obtain the total energy equation ( 22) by adding the averaged kinetic equation ( 8) from Lemma 1 to Equation ( 12) in Lemma 2, and applying the result from Lemma 3 to have\n\nwhere Ψ is defined in (12), and Q is defined in (17). The potential equation, on the other hand, is given by Equation ( 9), in Lemma 1. Then, combining ( 22) and ( 9), we obtain the desired total energy equation (20).\n\nNote that the total energy function E in Theorem 1 acts as a convex mathematical entropy function of system (3) with corresponding entropy flux F, where the convexity can be readily verified by computing the Hessian of E with respect to the conservative variables u. The next corollary introduces the entropy variables w related to (3).\n\nCorollary 1. The entropy variables, denoted by w 1, w 2,..., w N+2, associated to the SWME system (3) are:\n\nfor all i = 1,..., N.\n\nProof. The entropy variables are defined as the partial derivative of the entropy function E with respect to the conservative variables (h, hu m, hα). Therefore computing\n\nRemark 1. Notice that the energy function is independent of the coefficients A i jk and B i jk. Thus, the entropy function and entropy variables corresponding to the linearized model SWLME (6) are the function E in (20) and vector w from Corollary 1, respectively. The total flux, which also corresponds to the entropy flux for the SWLME and was first derived in [20] is, on the other hand, given by\n\nIn the next section, we discuss the case of including the friction effect, which turns out to be a source term in the model equations, and establish that two widely used friction terms described in the literature are indeed entropy dissipative with respect to the developed entropy variables.\n\n$$u(x, ζ, t) = u m (x, t) + N i=1 α i (x, t)ϕ i (ζ),(1)$$\n\n$$ζ = (z -b)/h ∈ [0, 1]$$\n\n$$L i = L i (ζ) defined in the interval [-1, 1], by ϕ i (ζ) (-1) i L i (2ζ -1), for i = 1,..., N, 0 ≤ ζ ≤ 1.(2)$$\n\n$$ϕ 1 (ζ) = 1 -2ζ, ϕ 2 (ζ) = 6ζ 2 -6ζ + 1, ϕ 3 (ζ) = -20ζ 3 + 30ζ 2 -12ζ + 1.$$\n\n$$∂ t h + ∂ x (hu m ) = 0,(3a)$$\n\n$$∂ t (hu m ) + ∂ x hu 2 m + Ψ + 1 2 gh 2 = -gh∂ x b,(3b)$$\n\n$$∂ t (hα i ) + ∂ x 2hu m α i + A i = u m ∂ x (hα i ) -B i,(3c)$$\n\n$$Ψ h N i=1 1 2i + 1 α 2 i = h 1 3 α 2 1 + 1 5 α 2 2 + • • • + 1 2N+1 α 2 N.(4)$$\n\n$$A i h N j,k=1 A i jk α j α k, B i N j,k=1 B i jk α k ∂ x hα j,$$\n\n$$A i jk = (2i + 1) 1 0 ϕ i (ζ)ϕ j (ζ)ϕ k (ζ) dζ, B i jk = (2i + 1) 1 0 ϕ ′ i (ζ) ζ 0 ϕ j (s)ds ϕ k (ζ) dζ. (5$$\n\n$$)$$\n\n$$∂ t (hα) + ∂ x 2hu m α + A = u m ∂ x (hα) -B.$$\n\n$$∂ t h + ∂ x (hu m ) = 0, (6a$$\n\n$$)$$\n\n$$∂ t (hu m ) + ∂ x hu 2 m + Ψ + 1 2 gh 2 = -gh∂ x b,(6b)$$\n\n$$∂ t (hα) + ∂ x (2hu m α) = u m ∂ x (hα).(6c)$$\n\n$$λ 1,2 = u m ± gh + N i=1 3α 2 i 2i + 1 1/2, λ i+2 = u m, for i = 1,..., N.(7)$$\n\n$$∂ t hu 2 m 2 + ∂ x hu 3 m 2 + ghu m ∂ x (h + b) + u m ∂ x Ψ = 0,(8)$$\n\n$$∂ t g h 2 2 + ghb + g(h + b)∂ x (hu m ) = 0,(9)$$\n\n$$h∂ t u m + hu m ∂ x u m + gh∂ x (h + b) + ∂ x Ψ = 0. (10$$\n\n$$)$$\n\n$$1 2 ∂ t (hu m ) + h∂ t u m + 1 2 ∂ x hu 2 m + hu m ∂ x u m + gh∂ x (h + b) + ∂ x Ψ = 0,(11)$$\n\n$$∂ t Ψ 2 + ∂ x u m Ψ 2 + Ψ∂ x u m + N i=1 1 2i + 1 α i ∂ x A i + B i = 0. (12$$\n\n$$)$$\n\n$$h∂ t α i + h∂ x (u m α i ) + ∂ x A i + B i = 0, for i = 1,..., N.(13)$$\n\n$$∂ t       h α 2 i 2       + ∂ x       hu m α 2 i 2       + α 2 i h∂ x u m + α i ∂ x A i + B i = 0, (14$$\n\n$$)$$\n\n$$α i h∂ x (u m α i ) + α 2 i 2 ∂ x (hu m ) = ∂ x       hu m α 2 i 2       + α 2 i h∂ x u m.$$\n\n$$Q N i=1 1 2i + 1 α i ∂ x A i + B i = N i, j,k=1 A i jk + B i jk α i α k ∂ x (hα j ) + A i jk hα i α j ∂ x α k,(15)$$\n\n$$Q(x, t) = ∂ x Q(x, t),(16)$$\n\n$$Q(x, t) N i, j,k=1 A i jk + B i jk h(x, t)α i (x, t)α j (x, t)α k (x, t). (17$$\n\n$$)$$\n\n$$∂ x Q = N i, j,k=1 A i jk + B i jk α i α k ∂ x (hα j ) + A i jk + B i jk + A k ji + B k ji hα i α j ∂ x α k.$$\n\n$$B i jk + A k ji + B k ji = 0. (18$$\n\n$$)$$\n\n$$∂ t E(u) + ∂ x F(u) = 0, (19$$\n\n$$)$$\n\n$$E(u) hu 2 m 2 + h 2 N i=1 α 2 i 2i + 1 + g h 2 2 + ghb,(20)$$\n\n$$F(u) hu 3 m 2 + hu m 3 2 N i=1 α 2 i 2i + 1 + ghu m (h + b) + N i, j,k=1 1 2i + 1 A i jk + B i jk hα i α j α k.(21)$$\n\n$$∂ t hu 2 m 2 + Ψ 2 + ∂ x hu 3 m 2 + ghu m ∂ x (h + b) + ∂ x 3 2 u m Ψ + Q = 0, (22$$\n\n$$)$$\n\n$$w 1 = - u 2 m 2 - 1 2 N i=1 α 2 i 2i + 1 + g(h + b), w 2 = u m, w i+2 = α i 2i + 1,(23)$$\n\n$$w 1 = ∂ h E, w 2 = ∂ hu m E and w i+2 = ∂ hα i E for all i = 1,..., N.$$\n\n$$F LM (u) hu 3 m 2 + hu m 3 2 N i=1 α 2 i 2i + 1 + ghu m (h + b).$$\n\n## 2.1. Entropy dissipation for the friction term\n\nWhen friction terms are taken into account, the SWME (3) (or SWLME ( 6)) become non-homogeneous and a source term is included. For classical Newtonian slip friction, this source term is defined as [3] S Ns (u) -\n\nwhere 1 is the vector of ones in R N, vector r ∈ R N is defined componentwise, where each component is given by r i = 2i + 1 for all i = 1,..., N, and the matrix C = ( Ci j ) ∈ R N×N is defined for each component as\n\nThe constant λ > 0 is the so-called slip length, and ν > 0 is a viscosity related parameter. The first term in (24) corresponds to the bottom friction given by the slip condition, while the second term containing the C matrix is related to the bulk friction.\n\nIn the next lemma, we show that the friction term S Ns is defined is in fact entropy dissipative with respect to the entropy variables w defined in Corollary 1.\n\nLemma 4. The friction term S Ns defined by (24) is entropy dissipative with respect to w, namely\n\nProof. Multiplying the entropy vector w to the friction term S Ns we have\n\nwhich completes the proof.\n\nAn alternative friction term can be obtained by using a Manning law for the bottom friction [21], which is given by\n\nwhere n > 0 is called the Manning coefficient. In the next lemma, we show that S NM is also entropy dissipative.\n\nLemma 5. The friction term S NM defined by ( 28) is entropy dissipative with respect to w.\n\nProof. Multiplying the entropy vector w to the friction term S NM, we perform the same calculations done in the proof of Lemma 4 to obtain\n\nwhich completes the proof.\n\n$$ν λ u m + 1 T α           0 1 r           - ν h           0 0 Cα          , (24$$\n\n$$)$$\n\n$$Ci j (2i + 1) 1 0 ϕ ′ i (ζ)ϕ ′ j (ζ) dζ for i, j = 1,..., N.(25)$$\n\n$$w T S Ns (u) ≤ 0. (26$$\n\n$$)$$\n\n$$w T S Ns = - ν λ u m + 1 T α (u m + 1 T α) - ν h N i, j=1 1 0 ϕ ′ i (ζ)ϕ ′ j (ζ)dζ α i α j = - ν λ u m + 1 T α 2 - ν h 1 0 N i=1 ϕ ′ i (ζ)α i N j=1 ϕ ′ j (ζ)α j dζ = - ν λ u m + 1 T α 2 - ν h 1 0 N i=1 ϕ ′ i (ζ)α i 2 dζ ≤ 0,(27)$$\n\n$$S NM (u) - ρgn 2 h 1/3 u m + 1 T α u m + 1 T α           0 1 r           - ν h           0 0 Cα          ,(28)$$\n\n$$w T S NM = - ρgn 2 h 1/3 u m + 1 T α u m + 1 T α 2 - ν h 1 0 N i=1 ϕ ′ i (ζ)α i 2 dζ ≤ 0,(29)$$\n\n## 3. Numerical method\n\nBefore delving further into the numerical approximation of the SWME, we augment the system with the additional trivial evolution equation b t = 0. This reformulation incorporates the bottom topography source term into the nonconservative product, which simplifies the construction of well-balanced schemes [22]. To this end, we add the bottom topography to both vectors of conservative variables u = (h, hu m, hα 1, hα 2,..., hα N, b) T ∈ R N+3 as well as entropy variables w = (w 1, w 2,..., w N+2, b) T ∈ R N+3. The resulting augmented and non-homogeneous system (3) is then written in vector notation as follows\n\nwhere S = S(u) denotes a source term, which typically includes the friction related contributions, and\n\n, with 0 the (column) vector of zeros in R N, I the identity matrix in R N×N, and B(α) ∈ R N×N is the matrix whose elements are defined by\n\nThe first summand in the definition of B corresponds to the nonconservative coefficient matrix related to the linearized moment model SWLME, and is denoted by B LM while the second summand that involves the coefficients B i jk is denoted by BLM.\n\nTo evaluate the moment coefficients A i jk and B i jk in (5), and friction coefficient Ci j in (25), we construct the shifted Legendre polynomials (2) from a recurrence relation and apply Legendre-Gauss quadrature for accurate integration, see Appendix Appendix B for details. Furthermore, we introduce the standard jump and average operators, denoted by • and {{•}}, respectively, which for a given function φ = φ(x) are defined by\n\nIn the following sections, we first introduce an entropy conservative numerical scheme that recovers an integral version of the entropy conservation law (19) in the semi-discrete case. This is achieved by formulating a numerical scheme that satisfies a summation-by-parts (SBP) property and the derivation of entropy conservative numerical fluxes to recover a semi-discrete version of the relation\n\nThis entropy conservative scheme acts as a baseline that is further extended to one that is entropy stable, by adding suitable numerical dissipation at interfaces, in the sense that the numerical method satisfies the entropy inequality\n\nwhich accounts for entropy dissipation at discontinuities.\n\n$$∂ t u + ∂ x f(u) + B(u)∂ x u = S(u),(30)$$\n\n$$f(u) =               hu m hu 2 m + Ψ 2hu m α + A 0              , B(u) =                0 0 0 T 0 gh 0 0 T gh 0 0 -u m I 0 0 0 0 T 0                B LM (u) +                0 0 0 T 0 0 0 0 T 0 0 0 B(α) 0 0 0 0 T 0                BLM (u)$$\n\n$$B i j (α) = B i j1 α 1 + B i j2 α 2 + • • • + B i jN α N.$$\n\n$$φ := φ (x) = φ + -φ -, {{φ}} := {{φ}} (x) = 1 2 (φ + + φ -), where φ + = φ(x + ) and φ -= φ(x -) for all x ∈ Ω.$$\n\n$$Ω w T ∂ t u + ∂ x f(u) + B(u)∂ x u dx = Ω ∂ t E(u) + ∂ x F(u) dx = 0.$$\n\n$$Ω ∂ t E(u) + ∂ x F(u) dx ≤ 0,(31)$$\n\n## 3.1. Entropy conservative discontinuous Galerkin scheme\n\nFor the spatial approximation, we formulate a collocated nodal discontinuous Galerkin spectral element method (DGSEM) on Legendre-Gauss-Lobatto (LGL) nodes [23,24]. To account for the nonconservative terms of the system, the method is written in terms of fluctuations as introduced in [25,26] using the flux differencing volume integral from [17]. To apply this method, we consider a uniform discretization of the domain Ω into k = 1,..., K non-overlapping elements of size ∆x k. Each element is mapped onto the reference domain E = {ξ ∈ R : |ξ| ≤ 1}, where the solution is approximated in the space of polynomials up to degree P\n\nspanned by nodal Lagrange basis functions {l j (ξ)} P j=0 with interpolation nodes located LGL points and test functions are taken from the basis. We further introduce the discrete derivative operator D ∈ R P×P, that is obtained from the derivatives of the Lagrange basis functions, and approximate integrals with LGL quadrature formulas, with corresponding LGL quadrature weights {ω j } P j=0. This choice of discrete operators collocates the interpolation and quadrature nodes and ensures that the method is endowed with a diagonal norm summation-by-parts property [27], which is key to ensure entropy stability. The semi-discrete nonconservative DGSEM in flux-differencing form for the degree of freedom i = 0, 1,..., P is then given by [17]\n\nwhere δ i0 and δ iP denote the Kronecker delta for the indices 0 and P, respectively, and D ± are fluctuations in the context of path-conservative schemes as introduced in [28], satisfying the consistency condition D ± (u, u) = 0 and a path-conservation condition\n\nwhere Φ(s) Φ(s; u -, u + ) is a parametrization of the path connecting the states u -and u +. In the present work, we consider fluctuations that are derived by taking the linear path Φ(s) = u -+ s u and approximating the path integral in (33) with the trapezoidal rule to obtain\n\nwhere f * denotes a numerical flux function that is consistent with the physical flux f * (u, u) = f(u). As shown in [17, Lemma 2] we can construct an entropy conservative (EC) scheme from (32), i.e., a method that recovers a semidiscrete version of the entropy conservation law (19), if the fluctuations (34) are constructed from an EC numerical flux f EC := f EC (u -, u + ) that satisfies the entropy conservation condition\n\nwhere the term on the right-hand side of ( 35) is the entropy potential. Accordingly, we define the EC fluctuations\n\nIn the following section, we demonstrate how to construct such an EC numerical flux f EC for the SWME and SWLME. In this procedure we will exploit the hierarchical character of the SMWE to decompose the system into components that satisfy distinct entropy conservation conditions.\n\n$$u ≈ U k (ξ) = P j=0 U k j l j (ξ),$$\n\n$$ω i ∆x k 2 ∂ t U k i + ω i P m=0 2D im D -(U k i, U k m ) + δ i0 D + (U k-1 P, U k 0 ) + δ iP D -(U k P, U k+1 0 ) = 0,(32)$$\n\n$$D -(u -, u + ) + D + (u -, u + ) = f + 1 0 B(Φ(s))∂ s Φ(s) ds,(33)$$\n\n$$D ± (u -, u + ) = 1 2 B(u ± ) u ± f(u ± ) -f * (u -, u + ),(34)$$\n\n$$w T D = F ⇔ w T f EC -w T B u = w T f -F,(35)$$\n\n$$D ± EC (u -, u + ) = f EC (u -, u + ) -f(u -) + 1 2 B(u -) u.(36)$$\n\n## 3.2. Entropy conservative flux for the SWME\n\nIn order to construct an EC flux that satisfies (35), we first observe that both the SWE and SWLME are part of the algebraic definition of the SWME. The can be expressed by defining their respective fluxes as follows\n\nwhere the second summand in the definition of f LM will be denoted by fSWE. Then, we note that f LM (u) = f SWE (u) + fSWE (u). In addition, it is straightforward to write the flux corresponding of the SWME in (30) in terms of f LM with the equation f(u) = f LM (u) + fLM (u), where the additional flux contribution is defined by fLM (u) (0, 0, A T, 0) T. In a similar fashion, the nonconservative term exhibit the same hierarchical structure\n\nwhere Θ is the zero matrix in R N×N, and the second summand in the definition of B LM is denoted by BSWE. Then, the nonconservative term related to the SWME in (30) satisfies that B(u) = B LM (u) + BLM (u). The hierarchical property of the moment models allows us to compute the potentials as follows\n\nwhere F LM is entropy flux of the SWLME defined as the entropy flux F in (21) without the terms related to A i jk and B i jk, and the remaining flux is F LM F -F LM. Hence, we compute the entropy potentials\n\nFinally, the entropy potential related to the SWME given in ( 35) is given by\n\nWith the entropy potentials determined, we are equipped to derive the EC flux for the SWME system (3), i.e., a numerical flux that satisfies (35). To this end, we will make use of the hierarchical property of the SWME system. From [16], we know that a suitable entropy conservative flux for the SWE is the following\n\nwhich together with the nonconservative matrix B SWE defined in (38), and the flux f EC,SWE satisfies the following equation:\n\nNext, we discretize the remaining fluxes fSWE and fLM by\n\nwhere for all i, j = 1,..., N, the vector A jk = (A 1 jk, A 2 jk,..., A N jk ) T is in R N. We are now able to define our entropy conservative numerical flux for the SWME system (30)\n\nThe following theorem shows that the numerical flux f EC defined by Equation ( 46) is entropy conservative, i.e., it satisfies the entropy condition (35).\n\nTheorem 2. The numerical flux f EC defined in (46), which approximates the flux f in system (30) fulfills the entropy condition (35) with the entropy flux F defined in (21).\n\nProof. In the following, we make use of the hierarchical property of the SWME which is described in the previous section. Thanks to the linearity of the average operator {{•}} and the definition of f EC and B as sums of fluxes and nonconservative matrices, respectively, the left-hand side of Equation ( 35) can be written as\n\nThen, we compute the terms related to the shallow water equations as follows:\n\nwhere P LM is defined in (41), and we have used the identity 1 2 α 2 j {{hu m }} = α j {{hu m }} α j.\n\nOn the other hand, for the linearized moment model related terms in the right-hand side of (47), we use of the identity for the coefficients A i jk and B i jk given in (18), to obtain\n\nwhere P LM is defined in (41). Thus, replacing (48) and (49) into (47) and using Equation (39), we conclude that\n\nwhich concludes the proof.\n\n$$f SWE (u)               hu m hu 2 m 0 0              , f LM (u) f SWE (u) +               0 Ψ 2hu m α 0              , (37$$\n\n$$)$$\n\n$$B SWE (u)                0 0 0 T 0 gh 0 0 T gh 0 0 Θ 0 0 0 0 T 0               , B LM (u) = B SWE (u) +                0 0 0 T 0 0 0 0 T 0 0 0 -u m I 0 0 0 0 T 0               , (38$$\n\n$$)$$\n\n$$w T f -F = w T (f LM + fLM ) -F LM = w T f LM -F LM + w T fLM -F LM,(39)$$\n\n$$P LM = w T f LM -F LM = g(h + b) -1 2 u 2 m -1 2 Ψ/h hu m + u m hu 2 m + Ψ + 2u m Ψ -F LM = g(h + b)hu m + 1 2 hu 3 m + 5 2 u m Ψ -F LM = u m Ψ,(40)$$\n\n$$P LM = w T fLM -F LM = N i=1 α i A i 2i + 1 -F LM = - N i, j,k=1 B i jk 2i + 1 hα i α j α k.(41)$$\n\n$$w T f -F = P LM + P LM = u m Ψ - N i, j,k=1 B i jk 2i + 1 hα i α j α k. (42$$\n\n$$)$$\n\n$$f EC,SWE (u -, u + ) =               {{hu m }} {{hu m }} {{u m }} 0 0              ,(43)$$\n\n$$w T f EC,SWE -w T B SWE u = 0. (44$$\n\n$$)$$\n\n$$fEC,SWE (u -, u + )                     0 N j=1 {{hαj}}{{αj}} 2 j+1 {{hu m }} {{α}} + {{hα}} {{u m }} 0                    , fEC,LM (u -, u + )                     0 0 N j,k=1 hα j {{α k }} A jk 0                    ,(45)$$\n\n$$f EC (u -, u + ) f EC,SWE (u -, u + ) + fEC,SWE (u -, u + ) + fEC,LM (u -, u + ). (46$$\n\n$$)$$\n\n$$w T f EC -w T B u = w T f EC,SWE + fEC,SWE + fEC,LM -w T B SWE + BSWE + BLM u = w T fEC,SWE -w T BSWE u + w T fEC,LM -w T BLM u.(47)$$\n\n$$w T fEC,SWE -w T BSWE u = N j=1 1 2 j + 1 u m hα j α j + α j hα j {{u m }} + α j u m hα j = N j=1 1 2 j + 1 hu m α 2 j = P LM,(48)$$\n\n$$w T fEC,LM -w T BLM u = N i, j,k=1 Ãi jk α i hα j {{α k }} -Bi jk {{α i α k }} hα j = N i, j,k=1 -Bi jk + Bk ji α i hα j {{α k }} -Bi jk {{α i α k }} hα j = N i, j,k=1 -Bi jk α i hα j {{α k }} + α k hα j {{α i }} + {{α i α k }} hα j = N i, j,k=1 -Bi jk hα i α j α k = P LM,(49)$$\n\n$$w T f EC -w T B u = P LM + P LM = w T f -F,$$\n\n## 3.3. Entropy stable discontinuous Galerkin scheme\n\nThe result from Theorem 2 shows that we can construct an EC scheme if we introduce EC fluctuations (36) built from the EC flux (46) for both volume and surface contributions of the DGSEM (32). While the EC formulation is valid for smooth solutions, for discontinuous solutions entropy should be dissipated. So, instead, we require an entropy stable (ES) formulation that recovers the entropy inequality (31) discretely. To this end, we extend the EC fluctuation (36) to one that is ES by adding a modified version of the Rusanov dissipation, that is formulated in terms of entropy variables in [29,30]\n\nwhere |λ| max denotes the largest absolute eigenvalue from (7) and\n\nwith vectors y = (1, u m, α T ) T and z = (1, gh, 3gh,..., (2n + 1)gh) T is the positive definite inverse of the Hessian of the entropy function E (h, hu m, hα T ), evaluated at some intermediate state ū ∈ [u -, u + ], that for constant bottom topography satisfies the relation diag( H, 1) w = u.\n\nRemark 2. While the standard Rusanov dissipation is shown to be entropy dissipative, e.g., in [17,31], these results require a spatially convex entropy function which does not apply in the case of non-constant bottom topography.\n\nWith H positive definite it follows that 1 2 |λ| max w T Q ES w is non-negative and from [17, Theorem 4] the scheme\n\nfor the degree of freedom i = 0, 1,..., P, with ES surface fluctuations (36) and ES volume fluctuations (50) is ES, i.e., it recovers a semi-discrete integral version of the entropy inequality (31).\n\n$$D ± ES = D ± EC ± 1 2 |λ| max Q ES w = D ± EC ± 1 2 |λ| max H 0 0 0 w,(50)$$\n\n$$H := 1 g {{y}} {{y}} T + diag({{z}}),$$\n\n$$ω i ∆x k 2 ∂ t U k i + ω i P m=0 2D im D - EC U k i, U k m + δ i0 D + ES U k-1 P, U k 0 + δ iP D - ES U k P, U k+1 0 = 0,(51)$$\n\n## 3.4. Well-balanced approximation\n\nAnother important continuous property of both the SMWE (3) and SWMLE ( 6) is the existence of steady states, where the solution remains constant in time. For both kinds of models, several distinct steady state configurations are known and different approaches such as the global flux method [32], relaxation methods [20] or a formulation in terms of equilibrium variables [18] have demonstrated preservation for a general class of steady states. Specifically, one important steady state solution that we consider in this work is the so-called lake-at-rest steady state given by\n\nwhere H 0 denotes some constant water level. Preservation of this steady state solution in the discrete case, called wellbalancing, is an important aspect of any numerical method as many solutions represent small perturbations around this steady state and violations may introduce artificial waves on the other of the grid spacing [33]. The following Lemma demonstrates that our ES numerical scheme (51) exactly satisfies such a well-balanced property at the discrete level.\n\nLemma 6. The ES scheme (51) with fluctuations (50) and (36) constructed from the EC flux (46) is well-balanced for the lake-at-rest steady state, i.e., it exactly preserves solutions satisfying the lake-at-rest conditions (52).\n\nProof. We show that both volume and surface fluctuations in (51) vanish point-wise, when evaluated under lake-at-rest conditions (52). First consider the surface fluctuations (36). From the lake-at-rest (52) most components vanish as momenta and moments are set to zero due to vanishing velocities. The remaining contributions vanish as the total height h + b is constant D ± EC (u -, u + ) = 0, 1 2 gh ± h + b, 0,..., 0 T = 0, for all u -, u + ∈ U wb.\n\nThen, to show that the volume fluctuations (50) vanish, we only have to show that the additional dissipation term vanishes\n\n|λ| max H w = 0, for all u -, u + ∈ U wb, which follows directly as the entropy variables assume a constant value for the lake-at-rest conditions (52). Substituting both results into (51) shows that the time-derivative vanishes.\n\n$$U wb = u : h + b = H 0, hu m = 0, hα = 0, h > 0,(52)$$\n\n$$D ± ES (u -, u + ) = D ± EC (u -, u + ) ± 1 2$$\n\n## 4. Numerical examples\n\nIn this section, we present a set of numerical examples to verify our theoretical findings and demonstrate the performance of the proposed ES numerical scheme from Section 3. The numerical examples have been produced making use of the open-source framework Trixi.jl [34,35] for the semi-discretization. For the time integration, we employ a low storage five-stage fourth-order Runge-Kutta scheme by Carpenter and Kennedy [36] as implemented in DifferentialEquations.jl [37]. Unless the time step is specified, we use a CFL-based time step using the eigenvalues of the SWLME given by (7), which for the case of the SWME, only corresponds to an estimation.\n\nTo cover the cases in which the solution may feature shocks or high gradients, we supplement our ES scheme (51) in Section 3 with the subcell shock capturing approach developed in [38] using the indicator variable u 3 m. The shock capturing method is active in all examples with the exception of the accuracy test in Example 4.3, as it may affect the spatial order of accuracy.\n\nThe necessary code and instructions to reproduce the numerical results in this section are provided in a reproducibility repository [39].\n\n## 4.1. Example 1: Smooth wave with friction effect\n\nWith the purpose of comparing our numerical scheme with existing solvers for SWME, we consider the standard test example of a travelling water wave with periodic boundary conditions described in [3,Section 5]. In this example, the friction is modelled via the Newtonian slip law given by the function S = S Ns (24), which corresponds to the source term in (30). Furthermore, we set the slip length to λ = 0.1 and the friction constant to ν = 0.1, and use g = 1. The water body is initially described by a smooth bump given by the following height function\n\nwhile the bathymetry function for the flat bottom corresponds to b = 0. The initial average velocity and moments, for the modelling polynomial order N = 2, are given by u m (x, 0) = 0.25, α 1 (x, 0) = 0 and α 2 (x, 0) = -0.25\n\nFor the reference approximate solution, we use a finite volume polynomial viscosity based scheme (PVM) described in [8], in which system (30) is written in a fully non-conservative formulation combining the Jacobian of f and matrix B into a single system matrix. In addition, the viscosity matrix used is given by the HLL method, the path integrals are computed with linear paths and third order quadrature rule, time approximation is via forward Euler, and the Fortran 90 routines used can be found in [40].\n\nIn Figure 1, we present snapshots of the numerical solution at time t = 2 computed with the DG entropy preserving numerical scheme from Section 3 on a mesh with K = 256 elements varying the polynomial degree P. The reference solution, which is approximated with 2500 cells, corresponds to the black dashed line. In all components of the solution, the DG approximations tend to the reference solution as the polynomial degree P increases. We observe that already for P = 4, the numerical solution is comparable to the reference solution, even when the difference in the number of layers is large. Now, to visually compare our approximate solutions upon mesh refinement, we fix the polynomial degree to P = 2 and compute the solutions for K = 128, 256, 512. As before, all the components of the approximated solutions are compared to the reference curve in Figure 2 at time t = 2. As expected, the approximations converge to the reference curve, where the finer refinement for K = 512 closely matches the reference solution.\n\n$$h(x, 0) = 1 + exp 3 cos π(x + 1 2 ) -4 for x ∈ [-1, 1],$$\n\n$$for x ∈ [-1, 1].$$\n\n## 4.2. Example 2: Entropy dissipation of the friction terms\n\nWe now study the discrete entropy dissipation due to the Newtonian slip friction S = S Ns (24) and Newtonian Manning law S = S NM (28) described in Section 2.1. For this, we consider the same initial condition as in Example 1, with actual gravitational acceleration constant g = 9.81, and for S NM we use the Manning coefficient n = 0.0165 and constant density ρ = 1000. Furthermore, we use here the non-homogeneous version of the linearized model SWLME defined in (6). To measure the entropy dissipation in the entire domain Ω = [-1, 1] at each time point, we define the following quantities\n\nwhich are evaluated numerically using the polynomial approximation and quadrature formulas as described in Section 3. In Figure 3, we show the entropy dissipation at each time for the Newtonian slip D Ns, and Newtonian Manning case D NM varying the friction parameter ν. As shown in Section 2.1, in all cases and for both friction terms, the entropy dissipation remains negative and tends to zero as the time evolves. In both friction terms, the effect of larger values of ν is more pronounced at early times, where the entropy dissipation curves for ν = 1 are the ones of larger magnitude, with minima at t = 0. The small oscillations in both cases can be explained as a consequence of the periodic boundary conditions.   24) with λ = 0.1 and ν = 0.1, for different numbers of elements K. In all simulations, the polynomial degree is P = 2 and CFL = 0.9.\n\n$$D Ns (t) = 1 |Ω| Ω w T S Ns (u) dx, D NM (t) = 1 |Ω| Ω w T S NM (u) dx,$$\n\n## 4.3. Example 3: Accuracy tests\n\nIn order to determine the order of accuracy produced by our developed numerical scheme, we use as exact solution a manufactured solution, for which an additional source term needs to be supplemented to system (30). We let u ex = h ex, h ex u m,ex, h ex α T ex, b T be the smooth manufactured exact solution defined component-wise by\n\nfor all x ∈ 0, √ 2 = Ω and time t ≥ 0. The corresponding source term related to the chosen manufactured solution is then obtained by replacing u ex into the left-hand side of (30), that is\n\nWe let u n κ be the discrete solution computed with our numerical scheme at time t n with a fixed meshsize κ = ∆x = √ 2/2 l and l ∈ N, where the number of elements is K = 2 l. Then, to compute the numerical errors due to the spatial approximations, we compute the following L 2 -error of each of the unknown at a fixed time t n e i,κ  (24) and Newtonian Manning friction S NM (28) with respect to the entropy variables w defined in Corollary 1. In all simulations K = 256 and CFL = 0.9.  7  2.56e-08 3.99 2.15e-07 4.00 8.16e-09 5.12 1.04e-08 5.07 6.82e-10 4.00 2 8  1.59e-09 4.00 1.34e-08 4.00 1.02e-09 3.00 7.71e-10 3.76 4.26e-11 4.00 2 9  9.97e-11 with i = 1,..., N + 3, for a sequence of mesh refinements of K = 2 6,..., 2 10 elements. Note that since the bathymetry function b is constant in time, the numerical error related to this component is given by its L 2 -projection into the discontinuous Galerkin space. The rate of convergence between two consecutive values κ = √ 2/2 l-1 and κ = √ 2/2 l is defined as θ j,κ (t n ) = log(e n j,κ /e n j,κ )/log(2), for j = 1,..., N + 3.\n\n$$b(x) = 2 + 1 2 sin √ 2πx, h ex (x, t) = 7 + cos 2 √ 2πx cos 2πt -b(x, t), u m,ex (x, t) = 1 2, α i,ex (x, t) = 1 2 for i = 1,..., N,$$\n\n$$S ex (x, t) = ∂ t u ex + ∂ x f(u ex ) + B(u ex )∂ x u ex.$$\n\n$$(t n ) = ∥ u n κ -u ex (t n ) i ∥ L 2 (Ω),(53)$$\n\n## SWME\n\n(54) Furthermore, we fix the modelling order to N = 2 and use a fixed time step on each set of examples. In Table 1, we present the numerical errors and rates of convergence for the SWME ( 6) and SWLME (6) for polynomial degree P = 3 and ∆t = 10 -5, at final time t = 0.05. For both models, the results show that the numerical errors decrease with respect to the mesh refinement and the orders of convergence, for most of the components, tend to approximately P + 1, which is the expected order of convergence for a nodal discontinuous Galerkin approximation. Furthermore, we observe that although the approximation related to the computation of A i jk and B i jk is not in effect for the SWLME (bottom table), the errors related to h and u m do not differ significantly to the SWME (top table). In the particular case of the moment components hα 1 and hα 2, the errors in the SWLME are slightly larger than those for the SWME; however, the convergence rate are also higher.\n\n## 4.4. Example 4: Lake-at-rest test\n\nWe consider a test case to verify numerically that the scheme proposed in (51) preserves the lake-at-rest steady state (52) for the SWME as shown in Lemma 6. For this we consider the domain Ω = [-4, 4] with periodic boundary conditions on which we prescribe the following initial conditions for all x ∈ Ω. The functions in (55) represent a minor perturbation in the velocity and moments around a lake-at-rest with smooth bottom topography. We further set the gravitational acceleration g = 9.812 and consider the SWME with N = 2 moments. To obtain the numerical results, we discretize the domain into 64 equidistant elements and compute the solution up to a final time t = 8000 to examine long time behaviour.\n\nIn Figure 4, we present numerical results for polynomial degree P = 1 at final time, which are obtained with our proposed ES scheme (51) and a non well-balanced scheme, where we replaced the interface dissipation with the standard Rusanov dissipation term. Additionally, in Figure 5, we present time series data for the total entropy and the lake-at-rest error in the water height for the same configuration.\n\nFor the ES scheme, we observe that over time the initial perturbation in the velocity profile gets dissipated by the numerical dissipation and even after considerable time we recover the original steady state configuration, which demonstrates the well-balanced property of the scheme. For the non well-balanced scheme, on the other hand, we initially observe oscillations in the water height as is typical when using a non well-balanced scheme for the SWE. In Figure 5 this causes the first sudden rise in the lake-at-rest error, that then remains almost constant over considerate time. However, the initial moment perturbations then keep growing over time until the solution starts to converge towards a completely different solution, which becomes evident through a noticeable rise in both total entropy and lake-at-rest error shortly before t = 8000. At later times the solution then seemingly approaches a different steady state that is depicted in Figure 4.\n\nThe results verify the well-balanced property for our ES scheme as established in Lemma 6. Furthermore, the behavior for the non well-balanced scheme shows how violations of the entropy inequality and the well-balanced property may cause artificial oscillations and non-physical solutions.\n\n$$H(x, 0) = 1.75, b(x, 0) = e -x 2 /2, h(x, 0) = H(x, 0) -b(x, 0), u m (x, 0) = α 1 (x, 0) = α 2 (x, 0) =              -10 -3 for -1 ≤ x < 0, 10 -3 for 0 ≤ x ≤ 1, 0 for 1 < |x|,(55)$$\n\n## 5. Conclusions\n\nWe developed energy equations for the general shallow water moment equations (SWME) and the linearized shallow water moment equations (SWLME), and established the respective entropy variables. Notably, the derived energy function and entropy variables coincide for both models. We use the derived entropy variables to show that two of the standard models of friction used in moment models, the Newtonian slip and Newtonian Manning, are entropy dissipative.\n\nThe results of the continuous entropy analysis were then used to construct an entropy stable and well-balanced discontinuous Galerkin spectral element method (DGSEM) that recovers the energy equations for the SWME in the semi-discrete case. For the spatial approximation, an augmented system with trivial evolution equation for the bottom topography was introduced to incorporate the bottom topography source term into the nonconservative product. The proposed DGSEM was written in fluctuation form to account for the nonconservative products and a special fluxdifferencing volume integral in combination with entropy conservative numerical fluxes was introduced to ensure entropy conservation.\n\nFor the derivation of these entropy conservative numerical fluxes, a key factor was the hierarchical character of the SWME, which allowed us to derive the entropy conservative fluxes sequentially from those known for the classical shallow water equations [15]. A drawback of our constructed entropy conservative flux is that it is specifically tailored for path-integrals (related to non-conservative terms) approximated through trapezoidal rule in combination with linear paths.\n\nAn entropy stable variant of the proposed DGSEM, was then obtained by adding a modified Rusanov dissipation term, written in terms of entropy variables, to the fluctuations at element interfaces. We further demonstrated that the resulting method exactly preserves the lake-at-rest steady state. A shock capturing method in the line of [38] is incorporated as an additional functionality to improve the robustness in the presence of discontinuities. Finally, the theoretical findings and the performance of the numerical method were demonstrated in a series of numerical test cases.\n\nExtensions of this work can be conducted in a variety of research directions. A direct generalization of the total energy of the system and entropy variables is to consider domains in two horizontal spatial dimensions. The models here presented can also be extended to the non-hydrostatic case or coupled to the Exner equations for sediment transport. A detailed study regarding entropy dissipative models of friction for granular flows is also of interest. With respect to the numerical approximation, further studies can be conducted regarding the approximation of the path integrals related to the non-conservative terms and the corresponding construction of the entropy fluxes. More elaborated schemes, including extensions to positivity-preserving schemes that are well-balanced for general steadystate solutions are also of relevance. where we made use of Φ j (0) = 0 and Φ j (1) = 0 for all j = 1,..., N.\n\n## Appendix B. Computation of moment model components\n\nThe moment model introduces two tensors and one matrix built from the shifted Legendre basis functions. The shifted Legendre polynomials can be created from the Rodrigues' formula\n\nwhere ζ ∈ [0, 1]. However, for the purpose of computing quantities like B i jk, it is convenient to exploit that the Legendre polynomials and their derivatives can be constructed from recurrence relations. In particular, the shifted Legendre basis comes from the three term recurrence ϕ 0 = 1, ϕ 1 = 1 -2ζ, ϕ j = 2 j-1 j (1 -2ζ)ϕ j-1 -j-1 j ϕ j-2, for j = 2,..., N, and its derivatives come from the recurrence ϕ ′ 0 = 0, ϕ ′ 1 = -2, ϕ ′ j = ϕ ′ j-2 -2(2 j -1)ϕ j-1, for j = 2,..., N.\n\nWe first create the C matrix (25) needed to include friction in the moment model. We use the recurrence relation for the shifted Legendre polynomial derivatives to build the necessary components of the integral. The terms in the integrand ϕ ′ i and ϕ ′ j are polynomials of degree at most N -1. So, the highest degree of the integrand is a polynomial of degree (N -1) + (N -1) = 2N -2. We compute the integrals for the matrix C i j with i, j = 1,..., N, with Legendre-Gauss (LG) quadrature. This quadrature is defined on the interval [-1, 1], so we use the affine map ϕ ′ i (ξ)ϕ ′ j (ξ) dξ, where i, j = 1,..., N. By design, LG quadrature with M + 1 quadrature points is exact for polynomials up to degree 2M + 1. Thus, we select M = ⌈(2N -3)/2 + 1⌉ nodes to guarantee that the quadrature is exact for the construction of the C matrix. Next, we create the B i jk tensor (5) for the moment equations. We use the recurrence relations for the shifted Legendre polynomials and their derivatives to build the necessary components of the integral. In the integrand, we have ϕ ′ i which is a polynomial of degree at most N -1 and ϕ j and ϕ k are polynomials of degree N. Therefore, ζ 0 ϕ j (s) ds is a polynomial of degree N + 1 and the integrand for B i jk is, at most, a polynomial of degree (N -1) + (N + 1) + N = 3N. We, again, compute the integral with LG quadrature where we select M = ⌈(3N -1)/2 + 1⌉ nodes to ensure that the quadrature will be exact for the construction. As before, we use an affine map on the outer integral to have\n\nζ m 0 ϕ j (s) ds ϕ k (ξ) dξ,\n\n$$ϕ j (ζ) = 1 j! d j dx j ζ -ζ 2 j, for j = 1,..., N,$$\n\n$$B i jk = 2i + 1 2 1 -1 ϕ ′ i (ξ)$$\n\n## References\n\n1. 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