88,431 research outputs found
l-Degree Turan Density
No full textLet H-n be a k-graph on n vertices. For 0 <= l < k and an l-subset T of V (H-n), define the degree deg(T) of T to be the number of (k - l)-subsets S such that S boolean OR T is an edge in H-n. Let the minimum l-degree of H-n be delta(l) (H-n) = min{deg(T) : T subset of V (H-n) and vertical bar T vertical bar = l}. Given a family F of k-graphs, the l-degree Turan number ex(l) (n, F) is the largest delta(l) (H-n) over all F-free k-graphs H-n on n vertices. Hence, ex(0) (n, F) is the Turan number. We define l-degree Turan density to be pi(kappa)(l) (F) = lim sup(n ->infinity) ex(l)(n, F)/kappa(n-l). In this paper, we show that for k > l > 1, the set of pi(kappa)(l) (F) is dense in the interval [0, 1). Hence, there is no "jump" for l-degree Turan density when k > l > 1. We also give a lower bound on pi(kappa)(l) (F) in terms of an ordinary Turan density
Some exact results for generalized Turan problems
No full textFix a k-chromatic graph F. In this paper we consider the question to determine for which graphs H does the Turan graph Tk-1(n) have the maximum number of copies of H among all n-vertex F-free graphs (for n large enough). We say that such a graph H is F-Turan-good. In addition to some general results, we give (among others) the following concrete results: (i) For every complete multipartite graph H, there is k large enough such that H is K-k-Turan-good. (ii) The path P-3 is F-Turan-good for F with chi(F) >= 4. (iii) The path P-4 and cycle C-4 are C5-Turan-good. (iv) The cycle C-4 is F-2-Turan-good where F-2 is the graph of two triangles sharing exactly one vertex
Generalized rainbow Turan problems
No full textAlon and Shikhelman [J. Comb. Theory, B. 121 (2016)] initiated the systematic study of the following generalized Turan problem: for fixed graphs H and F and an integer n, what is the maximum number of copies of H in an n-vertex F-free graph?An edge-colored graph is called rainbow if all its edges have different colors. The rainbow Turan number of F is defined as the maximum number of edges in a properly edge-colored graph on n vertices with no rainbow copy of F. The study of rainbow Turan problems was initiated by Keevash, Mubayi, Sudakov and Verstraete [Comb. Probab. Comput. 16 (2007)].Motivated by the above problems, we study the following problem: What is the maximum number of copies of F in a properly edge-colored graph on n vertices without a rainbow copy of F? We establish several results, including when F is a path, cycle or tree
Rainbow Turan Methods for Trees
No full textThe rainbow Turan number, a natural extension of the well-studied traditionalTuran number, was introduced in 2007 by Keevash, Mubayi, Sudakov and Verstraete. The rainbow Tur ́an number of a graph F , ex*(n, F ), is the largest number of edges for an n vertex graph G that can be properly edge colored with no rainbow F subgraph. Chapter 1 of this dissertation gives relevant definitions and a brief history of extremal graph theory. Chapter 2 defines k-unique colorings and the related k-unique Turan number and provides preliminary results on this new variant. In Chapter 3, we explore the reduction method for finding upper bounds on rainbow Turan numbers and use this to inform results for the rainbow Turan numbers of specific families of trees. These results are used in Chapter 4 to prove that the rainbow Turan numbers of all trees are linear in n, which correlates to a well-known property of the traditional Turan numbers of trees. We discuss improvements to the constant term in Chapters 4 and 5, and conclude with a discussion on avenues for future work
On the Rainbow Turan number of paths
No full textLet F be a fixed graph. The rainbow Turan number of F is defined as the maximum number of edges in a graph on n vertices that has a proper edge-coloring with no rainbow copy of F (i.e., a copy of F all of whose edges have different colours). The systematic study of such problems was initiated by Keevash, Mubayi, Sudakov and Verstraete.In this paper, we show that the rainbow Turan number of a path with k + 1 edges is less than (9k/7 + 2) n, improving an earlier estimate of Johnston, Palmer and Sarkar.DC
On weighted Turan type inequality
No full textLet Hn be the class of real algebraic polynomials of degree n, whose zeros all lie in the interval [-1, 1]. Define || f || = max-1 ≤ x ≤ 1 | f(x)|. In 1939, Turan proved that for f ∈ Hn, || f ' || ≥ C√n || f ||. Usually, to establish weighted inequalities for polynomials requires more complicated techniques, especially in general cases. The object of this paper is to start the work in uniform norm under general requirements for weight functions
Turan H-densities for 3-graphs
No full textGiven an r-graph H on h vertices, and a family F of forbidden subgraphs, we define ex H (n, F) to be the maximum number of induced copies of H in an F-free r-graph on n vertices. Then the Turan H-density of F is the limit pi(H)(F) = (lim)(n ->infinity) ex(H)(n, F)/((n)(h)) This generalises the notions of Turan-density (when H is an r-edge), and inducibility (when F is empty). Although problems of this kind have received some attention, very few results are known. We use Razborov's semi-definite method to investigate Turan H-densities for 3-graphs. In particular, we show that pi(-)(K4)(K-4) = 16/27, with Turans construction being optimal. We prove a result in a similar flavour for K-5 and make a general conjecture on the value of pi(Kt)-(K-t). We also establish that pi(4.2)(empty set) = 3/4, where 4: 2 denotes the 3-graph on 4 vertices with exactly 2 edges. The lower bound in this case comes from a random geometric construction strikingly different from previous known extremal examples in 3-graph theory. We give a number of other results and conjectures for 3-graphs, and in addition consider the inducibility of certain directed graphs. Let (S) over right arrow (k) be the out-star on k vertices; i.e. the star on k vertices with all k 1 edges oriented away from the centre. We show that pi((S) over right arrow3)(empty set) = 2 root 3 - 3, with an iterated blow-up construction being extremal. This is related to a conjecture of Mubayi and Rodl on the Turan density of the 3-graph C-5. We also determine pi((S) over right arrowk) (empty set) when k = 4, 5, and conjecture its value for general k
Alburnoides velioglui Turan, Kaya, Ekmekçi & Doğan, 2014, sp. n.
No full textAlburnoides velioglui, sp. n. (Fig. 3) Holotype. FFR 0 1094, male, 79 mm SL; Turkey: Erzurum Prov.: Sırlı Stream, Euphrates River drainage; 40 ° 12 ’ 34 ’’N, 41 °04’00’’E, coll. D. Turan, Y. Saral and M. Çelik, 17 Apr. 2004. Paratypes. FFR 0 1043, 15, 52–88 mm SL; same data as holotype. – FFR 0 1036, 8, 30–83 mm SL; Turkey: Erzurum Prov.: Toprakkale Stream, Euphrates River drainage; D. Turan, Y. Saral and M. Çelik, 0 3 Aug. 2007. – FFR 0 1106, 20, 51–88 mm SL; Turkey: Malatya Prov.: Sultansuyu Stream, Euphrates River drainage; D. Turan, E. Doğan and C. Kaya, 19 Oct. 2013. Additional material (non types). FFR 0 1105, 7, 50–69 mm SL; Turkey: Erzurum Prov.: Karasu Stream, Euphrates River drainage; D. Turan, C. Kaya and E. Doğan, 8 Aug. 2013. – FFR 0 1096, 5, 58–67 mm SL; Turkey: Sivas Prov.: Divriği Stream, Euphrates River drainage; D. Turan, C. Kaya and E. Doğan, 10 Aug. 2013. – FFR 0 1029, 2, 67–85 mm SL; Turkey: Tunceli Prov.: Munzur Stream, Euphrates River drainage; D. Turan, Y. Saral and M. Çelik, 12 Aug. 2013. Diagnosis. Alburnoides velioglui is distinguished from all species of Alburnoides in Turkey and adjacent waters by the following combination of characters (none unique to the species): a poorly developed ventral keel between pelvic and anal fins, completely scaled; body depth at dorsal-fin origin 24–29 % SL; caudal-peduncle depth 10–12 % SL and 1.9–2.2 times in its length; predorsal length 48–55 % SL; mouth terminal, the tip of the mouth cleft between level of lower margin of pupil and lower margin of eye; the tip of upper lip not projecting beyond the lower lip (tip of both lips are equal) in most specimens; snout with rounded tip; dark grey stripe distinct on anterior and posterior parts of body; pigmentation of lateral-line distinct (Fig. 3); 45–53 + 1–2 lateral-line scales, 9–11 scale rows between lateral-line and dorsal-fin origin, 4–5 scale rows between lateral-line and anal-fin origin, 11 ½– 13 ½ branched anal-fin rays; pharyngeal teeth 5.1 – 2.4 or 5.2 – 2.4, markedly hooked; number of total vertebrae 41–42 with mode of 42 (including 4 Weberian vertebrae and last complex centrum), comprising 20–22 with mode of 21 abdominal, and 20–21 with mode of 21 caudal vertebrae. Description. General appearance is shown in Figure 3; morphometric and meristic data are given in Tables 1 and 2. Body moderately deep and slightly compressed laterally. Caudal-peduncle depth 1.9–2.2 times in its length. Dorsal profile slightly convex, ventral profile equal or less convex than dorsal profile. Predorsal length 1.8–2.1 times in SL. Prepelvic length 2.0– 2.2 times in SL. Head short, approximately 0.9 –1.0 times body depth at dorsal fin origin, dorsal profile slightly convex at interorbital area, markedly convex at snout. Snout somewhat short, with rounded tip, approximately equal to eye diameter and smaller than interorbital width. Mouth terminal, with slightly marked chin. The tip of the mouth cleft approximately on level of lower margin of pupil or slightly below. The ventral keel poorly developed, completely scaled. Lateral-line with 46 (1), 47 (3), 48 (5), 49 (6), 50 (5), 51 (1), 52 (1), 53 (2) or 55 (1) scales; 9 (4), 10 (15) or 11 (6) scales rows between lateral-line and dorsal-fin origin; 4 (10) or 5 (15) scales between lateral-line and anal-fin origin. Gill rakers 1–2 + 3–4 = 5–6 on first gill arch. Dorsal fin with 3 simple and 8 ½ (22) and 9 ½ (3) branched rays, outer margin straight or slightly convex, its origin in front of vertical at mid-point of pelvic-anal distance. Pectoral fin short, not reaching pelvic-fin origin, outer margin convex, with 1 simple and 12 (1), 13 (9) or 14 (15) branched rays. Pelvic-fin short, not reaching the origin of anal-fin but reaching anus, with 1 simple and 7 branched rays, outer margin convex. Anal fin slender, with 3 simple and 11 ½ (7), 12 ½ (16) or 13 (2) branched rays, outer margin slightly concave posteriorly. Caudal-fin moderately forked, lobes slightly rounded. Pharyngeal teeth 5.1 – 2.4 or 5.2 – 2.4, markedly hooked. Number of total vertebrae 41 (2) or 42 (18); predorsal vertebrae 13 (8), 14 (11) or 15 (1); number of abdominal vertebrae 20 (2), 21 (13) or 22 (5), and that of caudal vertebrae 20 (4) or 21 (16); the abdominal region equal or longer than the caudal region, and the difference between the abdominal and caudal numbers varies from + 2 to – 1; vertebral formulae 22 + 20 (4), 21 + 21 (14) or 20 + 21 (2). Its maximum known size is 88 mm SL. Sexual dimorphism. There are small tubercles on membrane of anal and pelvic fins in males. The length of the paired fins does not show a statistically significant difference between males and females as it often does in other Alburnoides species. Coloration. Formalin-preserved adults and juveniles brownish on back and upper part of flank, light brownish on lower part of flank and belly. Caudal and dorsal fins light grey; pectoral, pelvic and anal fins yellowish. Pigmentation of lateral-line is distinct on both anterior and posterior parts of body. There is a narrow dark grey stripe (its width smaller than eye diameter) on upper part of flank from posterior margin of operculum to caudal peduncle, distinct anteriorly and posteriorly. Distribution and notes on biology. Alburnoides velioglui is known only from the northern Euphrates drainage (Sırlı and Toprakkale streams [drainages of Karasu] and Karasu Stream]) (Fig. 1). It inhabits swift and clear flowing water with cobble and pebbles. Capoeta umbla (Heckel, 1843); Barbus lacerta Heckel, 1843; Alburnus mossulensis Heckel, 1843, Oxynoemacheilus sp., and Salmo sp. have been collected with A. velioglui. Etymology. The species is named for Hasan Basri Velioğlu, Medical Doctor, who eased and contributed to our earlier and present studies by radiography.Published as part of Turan, Davut, Kaya, Cüneyt, Ekmekçi, F. Güler & Doğan, Esra, 2014, Three new species of Alburnoides (Teleostei: Cyprinidae) from Euphrates River, Eastern Anatolia, Turkey, pp. 101-116 in Zootaxa 3754 (2) on pages 106-108, DOI: 10.11646/zootaxa.3754.2.1, http://zenodo.org/record/22799
Alburnoides kosswigi Turan, Kaya, Bayçelebi, Bektaş & Ekmekçi, 2017, new species
No full textAlburnoides kosswigi, new species (Fig. 5) Holotype. FFR 0 1064, 80 mm SL; Turkey: Kütahya prov.: stream Porsuk about 3 km south of Hacıazizler, 39°20'59"N, 30°02'17"E; D. Turan, C. Kaya and E. Bayçelebi, 17 Aug. 2014. Paratypes. FFR 0 1135, 39, 40–61 mm SL; same data as holotype. — FFR 0 1107, 25, 41–82 mm SL; Turkey: Eskişehir prov.: stream Porsuk about 7 km south of Sazak, 39°43'26"N, 31°37'09"E; D. Turan, C. Kaya and E. Bayçelebi, 0 4 Sep. 2012. — FFR 0 1133, 52, 48–97 mm SL; Turkey: Ankara prov.: stream Kirmir about 2 km south of Yeşilöz, 40°14'13"N, 32°15'43"E; D. Turan, C. Kaya and E. Bayçelebi, 4 Sep. 2014. — FFR 0 1060, 27, 33–72 mm SL; Turkey: Ankara prov.: stream İlhanlı about 2 km south of İlhanköy, 40°05'38"N, 32°14'53"E; D. Turan, 15 June 2005. Diagnosis. Alburnoides kosswigi is distinguished from A. fasciatus by having a scaleless keel between the posterior pelvic base and the anus, rarely with 1-3 scale covering the anterior part of the keel [Fig. 2 f] (vs. covered by (2) 3–5 scales [Fig. 2 b]). Alburnoides kosswigi has fewer branched anal-fin rays (11½–13½, rarely 14½ vs. 13½–15½) and fewer abdominal vertebrae (20–21 vs. vs. 21–22) than A. fasciatus. In Alburnoides kosswigi, the tip of the upper lip is slightly projecting beyond lower lip (vs. not projecting in A. fasciatus). Alburnoides kosswigi is distinguished from A. tzanevi by having a scaleless keel between the posterior pelvic base and the anus, rarely with 1-3 scale covering the anterior part of the keel [Fig. 2 f] (vs. covered by 5–7 scales [Fig. 2 c]). Also, A. kosswigi has a deeper body (body depth dorsal-fin origin 26–31% SL, mean 29 vs. 24–27, mean 25) and a wider interorbital distance (7–10% SL, mean 9 vs. 6–8, mean 7). In A. kosswigi, the interorbital distance is wider than the eye diameter and the snout length (vs. snout length equal to interorbital distance but smaller than eye diameter). Alburnoides kosswigi also has a deeper anal-fin (18–21% SL, 20.1 vs. 17–19, mean 17.7) and a longer pelvic-fin (16–20% SL, mean 18.1 vs. 14–17, mean 16.2). Alburnoides kosswigi is distinguished from A. manyasensis by having a scaleless keel between the posterior pelvic base and the anus, rarely with 1-3 scale covering the anterior part of the keel [Fig. 2 f] (vs. covered by (2) 3– 5 scales [Fig. 2 d]) and a more slender caudal peduncle (9–11% SL vs. 11–12). In A. kosswigi, the upper lip is slightly projecting beyond the lower lip in most specimens (vs. not projecting) and the interorbital distance is greater than eye diameter (vs. smaller than eye diameter). Alburnoides kosswigi is distinguished from A. kurui by having fewer gill rakers (5–7 vs. 7–9), fewer scale rows between the lateral-line and the anal-fin origin (3–5 vs. 5–7), fewer branched anal-fin rays (11–14½ vs. 13–15½), fewer total vertebrae (39–42 vs. 42–43) and fewer caudal vertebrae (19–21 vs. 21–22). In A. kosswigi, the eye diameter is greater than the snout length (vs. almost equal) and the body depth at the dorsal-fin origin is 0.9–1.0 times in the head length (vs. 0.8–0.9). Alburnoides kosswigi is distinguished from A. freyhofi by having a more slender body (body depth at dorsal-fin origin 0.9–1.0 times in head length vs. 0.8–0.9) and a longer caudal peduncle (17–22% SL vs. 14–18). Alburnoides kosswigi has fewer branched anal-fin rays (11–13½, rarely 14½ vs. 14–16½) and a shorter anal-fin base (16–20% SL vs. 20–25). In A. kosswigi, the upper lip is slightly projecting beyond the lower lip in most individuals (vs. not projecting). Description. For general appearance see Fig. 5. Morphometric and meristic data are provided in Table 2 –3. Body moderately deep, slightly compressed laterally. Dorsal profile slightly convex, ventral profile equal or less convex than dorsal profile. Distance between pelvic-fin origin and anal-fin origin 1.2–1.4 times in distance between pectoral-fin origin and pelvic-fin origin. Head 0.9–1.0 times in body depth at dorsal-fin origin, interorbital area and snout slightly convex. Caudal-peduncle depth 1.8–2.2 times in its length. Snout short and with slightly pointed tip, snout length smaller than both eye diameter and interorbital distance. Mouth terminal in most individuals, without chin. Tip of upper lip slightly projecting beyond lower lip in most individuals. Tip of mouth cleft approximately at level of middle of pupil or slightly above. Ventral keel well developed, scaleless or rarely 1– 3 scales covering anterior portion of keel (Fig. 2 f). Largest known specimen 97 mm SL. Lateral-line with 43–57 scales, 8–12 scale rows between lateral-line and dorsal-fin origin and 3–5 scale rows between lateral-line and anal-fin origin. Gill rakers 1–2 + 4–6 = 5–7 on outer side of first gill arch. Dorsal fin with 3 simple and 8½–9½ branched rays, outer margin straight. Pectoral fin with 1 simple and 13–15 branched rays, outer margin slightly convex or convex. Pelvic fin with 1 simple and 6–7 branched rays, outer margin convex. Anal fin with 3 simple and 11½–13½ (14½) branched rays, outer margin concave posteriorly. Caudal-fin moderately forked, lobes slightly rounded. Pharyngeal teeth 2.5–4.2, markedly hooked at tip not serrated. Total vertebrae 39 (1), 40 (15), 41 (8) and 42 (4), predorsal vertebrae 13 (13) and 14 (15), abdominal vertebrae 20 (13) and 21 (15) and caudal vertebrae 19 (2), 20 (22) and 21 (4). Abdominal region equal or longer than caudal region, and difference between abdominal and caudal numbers varies from zero to +1. Vertebral formulae 20+19 (1), 20+20 (13), 21+20 (11) and 21+21 (3). Sexual dimorphism. Male with small tubercles on scales, head and rays of anal and dorsal fins. Coloration. Formalin-preserved individuals brownish on back and upper part of flank, light yellowish on lower part of flank and belly. Spots along lateral line above and below pores distinct in anterior part of flank, indistinct on caudal peduncle in most individuals. A distinct narrow black stripe on upper part of flank from posterior margin of operculum to caudal peduncle, its width smaller than eye diameter. Very few pigment cells on pockets of flank scale in most individuals. Caudal, dorsal and anal fins grey; pectoral and pelvic fins yellowish. Numerous black pigment cells on all fin rays. Distribution and notes on biology. Alburnoides kosswigi is known from the Sakarya River drainage. See Fig. 1 for a map of the findings of this species. It inhabits swift and clear flowing water with cobble and pebbles. Etymology. The species is named for Curt Kosswig (Hamburg & Istanbul), the father of ichthyology in Turkey. A noun in genitive, indeclinable.Published as part of Turan, Davut, Kaya, Cüneyt, Bayçelebi, Esra, Bektaş, Yusuf & Ekmekçi, F. Güler, 2017, Three new species of Alburnoides from the southern Black Sea basin (Teleostei: Cyprinidae), pp. 565-577 in Zootaxa 4242 (3) on pages 573-575, DOI: 10.11646/zootaxa.4242.3.8, http://zenodo.org/record/37697
Turan type inequalities for confluent hypergeometric functions of the second kind
No full textIn this paper we deduce some tight Turan type inequalities for Tricomi confluent hypergeometric functions of the second kind, which in some cases improve the existing results in the literature. We also give alternative proofs for some already established Turan type inequalities. Moreover, by using these Turan type inequalities, we deduce some new inequalities for Tricomi confluent hypergeometric functions of the second kind. The key tool in the proof of the Turan type inequalities is an integral representation for a quotient
of Tricomi confluent hypergeometric functions, which arises in the study of the infinite divisibility of the Fisher-Snedecor F distribution
- …
