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Applied mathematics and nonlinear sciences in the war on cancer

., (2015), Consensus recommendations for a standardized Brain Tumor Imaging Protocol in clinical trials , Neuro-Oncology, 17, No 9, 1188-1198. 10.1093/neuonc/nov095 Ellingson B. M. 2015 Consensus recommendations for a standardized Brain Tumor Imaging Protocol in clinical trials Neuro-Oncology 17 9 1188 1198 10.1093/neuonc/nov095 [55] N. Gordillo, E. Montseny and P. Sobrevilla, (2013), State of the art survey on MRI brain tumor segmentation , Magnetic Resonance Imaging, 31, No 8, 1426-1438. 10.1016/j.mri.2013.05.002 Gordillo N. Montseny E. Sobrevilla P. 2013 State

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A Note on Upper Bounds for Some Generalized Folkman Numbers

). [5] N. Kolev, A multiplicative inequality for vertex Folkman numbers , Discrete Math. 308 (2008) 4263–4266. doi:10.1016/j.disc.2007.08.008 [6] Y. Li and Q. Lin, On generalized Folkman numbers , Taiwanese J. Math. 21 (2017) 1–9. doi:10.11650/tjm.21.2017.7710 [7] Q. Lin and Y. Li, A Folkman linear family , SIAM J. Discrete Math. 29 (2015) 1988–1998. doi:10.1137/130947647 [8] N. Nenov, An example of a 15 -vertex (3, 3) -Ramsey graph with clique number 4, C.R. Acad. Bulgare Sci. 34 (1981) 1487–1489, in Russian. [9] J. Nešetřil and

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A Note on Lower Bounds for Induced Ramsey Numbers

Abstract

We say that a graph F strongly arrows a pair of graphs (G,H) and write F ind(G,H) if any 2-coloring of its edges with red and blue leads to either a red G or a blue H appearing as induced subgraphs of F. The induced Ramsey number, IR(G,H) is defined as min{|V (F)| : F ind (G,H)}. We will consider two aspects of induced Ramsey numbers. Firstly we will show that the lower bound of the induced Ramsey number for a connected graph G with independence number α and a graph H with clique number ω is roughly ω2α2. This bound is sharp. Moreover we will also consider the case when G is not connected providing also a sharp lower bound which is linear in both parameters.

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Ramsey Properties of Random Graphs and Folkman Numbers

Abstract

For two graphs, G and F, and an integer r ≥ 2 we write G → (F)r if every r-coloring of the edges of G results in a monochromatic copy of F. In 1995, the first two authors established a threshold edge probability for the Ramsey property G(n, p) → (F)r, where G(n, p) is a random graph obtained by including each edge of the complete graph on n vertices, independently, with probability p. The original proof was based on the regularity lemma of Szemerédi and this led to tower-type dependencies between the involved parameters. Here, for r = 2, we provide a self-contained proof of a quantitative version of the Ramsey threshold theorem with only double exponential dependencies between the constants. As a corollary we obtain a double exponential upper bound on the 2-color Folkman numbers. By a different proof technique, a similar result was obtained independently by Conlon and Gowers.

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Antipodal Edge-Colorings of Hypercubes

References [1] M. DeVos and S. Norine, Edge-antipodal colorings of cubes, The Open Problem Garden. http://garden.irmacs.sfu.ca/?q=op/edge antipodal colorings of cubes [2] T. Feder and C. Subi, On hypercube labellings and antipodal monochromatic paths, Discrete Appl. Math. 161 (2013) 1421-1426. doi: 10.1016/j.dam.2012.12.025 [3] K. Gandhi, Maximal monochromatic geodesics in an antipodal coloring of hypercube (2015), manuscript. http://math.mit.edu/research/highschool/primes/materials/2014/Gandhi

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Another View of Bipartite Ramsey Numbers

R eferences [1] E. Andrews, G. Chartrand, C. Lumduanhom and P. Zhang, Stars and their k-Ramsey numbers , Graphs Combin. 33 (2017) 257–274. doi:10.1007/s00373-017-1756-9 [2] L.W. Beineke and A.J. Schwenk, On a bipartite form of the Ramsey problem , in: Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975) 17–22. [3] Z. Bi, G. Chartrand and P. Zhang, Party problems and Ramsey numbers , preprint. [4] G. Chartrand and P. Zhang, Chromatic Graph Theory (Chapman & Hall/CRC Press, Boca Raton, FL, 2009

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Bifurcation Analysis of Hysteretic Systems with Saddle Dynamics

two real equilibria of saddle type. After some preparation work, we get a canonical form which is general enough for our purpose. See the appendix for a derivation of the proposed canonical form. We deal with an upper system of saddle type { x ˙ = 2 γ ( x − x E ) − ( y − y E ) , y ˙ = ( γ 2 − 1 ) ( x − x E ) ,    x ⩽ 1 , $$ \begin{equation} \left\{\begin{array}{r c l} \dot{x}&=& 2\gamma (x-x_E)-(y-y_E),\\ \dot{y}&=&(\gamma^2-1)(x-x_E), \end

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A Note on the Ramsey Number of Even Wheels Versus Stars

. Combin. Math. Combin. Comput. 55 (2005) 123–128. [9] A. Korolova, Ramsey numbers of stars versus wheels of similar sizes , Discrete Math. 292 (2005) 107–117. doi:10.1016/j.disc.2004.12.003 [10] B. Li and I. Schiermeyer, On star-wheel Ramsey numbers , Graphs Combin. 32 (2016) 733–739. doi:10.1007/s00373-015-1594-6 [11] V. Rosta, On a Ramsey-type problem of J.A. Bondy and P. Erdös , II, J. Combin. Theory Ser. B 15 (1973) 105–120. doi:10.1016/0095-8956(73)90036-1 [12] Surahmat and E.T. Baskoro, On the Ramsey number of a path or a star

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On the Restricted Size Ramsey Number Involving a Path P 3

. Comput. 24 (2015) 551–555. doi:10.1017/S096354831400056X [9] P. Erdős, R.J. Faudree, C.C. Rousseau and R. Schelp, The size Ramsey number , Period. Math. Hungar. 9 (1978) 145–161. doi:10.1007/BF02018930 [10] P. Erdős and R.J. Faudree, Size Ramsey function , in: Sets, graphs, and numbers, Colloq. Math. Soc. Janos Bolyai 60 (1991) 219–238. [11] R.J. Faudree, C.C. Rousseau and J. Sheehan, A class of size Ramsey numbers involving stars , in: Graph Theory and Combinatorics - A Volume in Honor of Paul Erdős, B. Bollobás (Ed(s)), (Academic Press

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Characterizations of Generalized Exponential Trichotomies for Linear Discrete-time Systems

. Ducrot, P. Magal, O. Seydi, P ersistence of Exponential Trichotomy for Linear Operators: A Lyapunov-Perron Approach, J. Dynam. Differential Equations, Vol. 28 (2016), 93-126. [5] S. Elaydi, K. Janglajew, D ichotomy and Trichotomy of Difference Equations, J. Difference Equ. Appl. Vol. 3 (1998), 417-448. [6] N. Lupa, M. Megan, G eneralized exponential trichotomies for abstract evolution operators on the real line, J. Funct. Spaces Appl., Vol. 2013, Article ID 409049, 8 pages, 2013. [7] C.L. Mihit, M. Megan, T. Ceausu, T he Equivalence of Datko and

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