... Antoni Wójcik,1 Tomasz Łuczak,2,* Paweł Kurzyński,1 Andrzej Grudka,1 Tomasz Gdala,2 andMałgor... more ... Antoni Wójcik,1 Tomasz Łuczak,2,* Paweł Kurzyński,1 Andrzej Grudka,1 Tomasz Gdala,2 andMałgorzata Bednarska2 1Faculty of Physics, Adam Mickiewicz University, Umultowska 85 ... Ideally,such a network should meet both the sim-plicity and the minimal control requirements ...
Let (Q,): be a random Q"-process, that is let Q,, be the empty spanning subgraph of the cube Q" a... more Let (Q,): be a random Q"-process, that is let Q,, be the empty spanning subgraph of the cube Q" and, for 1 s t I M = n N / 2 = n2"-', let the graph Q, be obtained from Q,-, by the random addition of an edge of Q" not present in Q , -, . When t is about N / 2 , a typical Q, undergoes a certain "phase transition": the component structure changes in a sudden and surprising way. Let t = (1 + E ) N /~ where E is independent of n . Then all the components of a typical Q, have o ( N ) vertices if E < 0, while if E > O then, as proved by Ajtai, Komlbs, and Szemeridi, a typical Q, ha5 a "giant" component with at least ( Y ( E ) N vertices, where a ( € ) > 0. In this note we give essentially best possible results concerning the emergence of this giant component close to the time of phase transition. Our results imply that if 17 > 0 is fixed and t 5 (1 -n -" ) N / 2 , then all components of a typical Q, have at most n"") vertices, where p ( q ) > 0. More importantly, if 60(log n)'/n 5 E = .(a) = o(l), then the largest component of a typical Q, has about 2 e N vertices, while the second largest component has order O ( n E -2 ) . Loosely put, the evolution of a typical Q" process is such that shortly after time N / 2 the appearance of each new edge results in the giant component acquiring 4 new vertices.
We study the dynamics of a generalization of a quantum coin walk on the line, which is a natural ... more We study the dynamics of a generalization of a quantum coin walk on the line, which is a natural model for a diffusion modified by quantum or interference effects. In particular, our results provide surprisingly simple explanations for recurrence phenomena observed by Bouwmeester et al. [Phys. Rev. A 61, 13410 (1999)]] in their optical Galton board experiment, and a description of a stroboscopic quantum walk given by Buerschaper and Burnett [quant-ph/0406039] through numerical simulations. We also provide heuristic explanations for the behavior of our model which show, in particular, that its dynamics can be viewed as a discrete version of Bloch oscillations.
... Antoni Wójcik,1 Tomasz Łuczak,2,* Paweł Kurzyński,1 Andrzej Grudka,1 Tomasz Gdala,2 andMałgor... more ... Antoni Wójcik,1 Tomasz Łuczak,2,* Paweł Kurzyński,1 Andrzej Grudka,1 Tomasz Gdala,2 andMałgorzata Bednarska2 1Faculty of Physics, Adam Mickiewicz ... It is shown that in such a system under fairly general conditions, we can expect a nearly perfect transfer of states. ...
Let F k be the family of graphs G such that all sufficiently large k-connected claw-free graphs w... more Let F k be the family of graphs G such that all sufficiently large k-connected claw-free graphs which contain no induced copies of G are subpancyclic. We show that for every k ≥ 3 the family F 1 k is infinite and make the first step toward the complete characterization of the family F 3 . ᭧
We show that every 3-connected claw-free graph which contains no induced copy of P 11 is hamilton... more We show that every 3-connected claw-free graph which contains no induced copy of P 11 is hamiltonian. Since there exist non-hamiltonian 3-connected claw-free graphs without induced copies of P 12 this result is, in a way, best possible. ß
We characterize all pairs of connected graphs fX ; Y g such that each 3-connected fX ; Y g-free g... more We characterize all pairs of connected graphs fX ; Y g such that each 3-connected fX ; Y g-free graph is pancyclic. In particular, we show that if each of the graphs in such a pair fX ; Y g has at least four vertices, then one of them is the claw K 1;3 , while the other is a subgraph of one of six specified graphs. ß
n denote the 3-uniform tight cycle, that is the hypergraph with vertices v1, . . . , vn and edges... more n denote the 3-uniform tight cycle, that is the hypergraph with vertices v1, . . . , vn and edges v1v2v3, v2v3v4, . . . , vn−1vnv1, vnv1v2. We prove that the smallest integer N = N (n) for which every red-blue coloring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C (3) n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl.
We show that there is an absolute constant 5 > 0 such that the number of sum-free subsets of any ... more We show that there is an absolute constant 5 > 0 such that the number of sum-free subsets of any finite abelian group G is where v(G) is the number of even order components in the canonical decomposition of G into a direct sum of its cyclic subgroups, and the implicit constant in the O-sign is absolute.
Four concepts are considered for the trophic level of a species in a community food web. The long... more Four concepts are considered for the trophic level of a species in a community food web. The long-way-uplevel (or LU-level) of species A is the length of the longest simple food chain from a basal species (one with no prey in the web) to A. (A simple chain is a chain that does not pass through any given species more than once.) The short-way-up-level (SU-level) of species A is the length of the shortest chain from a basal species to A. The long-way-down-level (LD-ievel) of species A is the length of the longest simple chain from species A to a top species (one with no consumers in the web). The short-way-down-level (SD-level) of species A is the length of the shortest chain from species A to a top species. The stratigraphy of a web is the analogue for species of the 'pyramid of numbers' for individuals: it is the frequency distribution of species according to level. The LU-, SU-, LD-, and SD-stratigraphies of the seven webs in the Briand-Cohen collection with 30 or more trophic species reveal no species with LU-level or LD-level more than 6, no species with SU-level more than 3, and no species with SD-level more than 2. In all seven webs, SD-levels are stochastically less than SU-levels: species tend to be closer to a top predator than to a basal species. Two stochastic models of food web structure (the cascade model and the homogeneous superlinear model) correctly predict that 95% or more of species should have LU-level and LD-level in the range 0-6. The models also correctly predict some details of the distribution of species in the SU-and SD-stratigraphies, particularly the fraction of species in level 1. The models do not, in general, correctly predict the distribution of species within the range 0-6 of LU-levels and LD-levels.
Dobson conjectured that for every t and k a tree with k vertices can always be embedded into a gr... more Dobson conjectured that for every t and k a tree with k vertices can always be embedded into a graph of minimum degree max{ (T ); (k − 1)=t } and girth at least 2t + 1. We show that the conjecture holds for all except a ÿnite number of pairs (t; k).
Let D(G) be the smallest quantifier depth of a first-order formula which is true for a graph G bu... more Let D(G) be the smallest quantifier depth of a first-order formula which is true for a graph G but false for any other non-isomorphic graph. This can be viewed as a measure for the descriptive complexity of G in first-order logic. We show that almost surely D(G) = Θ( ln n ln ln n ), ...
... Antoni Wójcik,1 Tomasz Łuczak,2,* Paweł Kurzyński,1 Andrzej Grudka,1 Tomasz Gdala,2 andMałgor... more ... Antoni Wójcik,1 Tomasz Łuczak,2,* Paweł Kurzyński,1 Andrzej Grudka,1 Tomasz Gdala,2 andMałgorzata Bednarska2 1Faculty of Physics, Adam Mickiewicz University, Umultowska 85 ... Ideally,such a network should meet both the sim-plicity and the minimal control requirements ...
Let (Q,): be a random Q"-process, that is let Q,, be the empty spanning subgraph of the cube Q" a... more Let (Q,): be a random Q"-process, that is let Q,, be the empty spanning subgraph of the cube Q" and, for 1 s t I M = n N / 2 = n2"-', let the graph Q, be obtained from Q,-, by the random addition of an edge of Q" not present in Q , -, . When t is about N / 2 , a typical Q, undergoes a certain "phase transition": the component structure changes in a sudden and surprising way. Let t = (1 + E ) N /~ where E is independent of n . Then all the components of a typical Q, have o ( N ) vertices if E < 0, while if E > O then, as proved by Ajtai, Komlbs, and Szemeridi, a typical Q, ha5 a "giant" component with at least ( Y ( E ) N vertices, where a ( € ) > 0. In this note we give essentially best possible results concerning the emergence of this giant component close to the time of phase transition. Our results imply that if 17 > 0 is fixed and t 5 (1 -n -" ) N / 2 , then all components of a typical Q, have at most n"") vertices, where p ( q ) > 0. More importantly, if 60(log n)'/n 5 E = .(a) = o(l), then the largest component of a typical Q, has about 2 e N vertices, while the second largest component has order O ( n E -2 ) . Loosely put, the evolution of a typical Q" process is such that shortly after time N / 2 the appearance of each new edge results in the giant component acquiring 4 new vertices.
We study the dynamics of a generalization of a quantum coin walk on the line, which is a natural ... more We study the dynamics of a generalization of a quantum coin walk on the line, which is a natural model for a diffusion modified by quantum or interference effects. In particular, our results provide surprisingly simple explanations for recurrence phenomena observed by Bouwmeester et al. [Phys. Rev. A 61, 13410 (1999)]] in their optical Galton board experiment, and a description of a stroboscopic quantum walk given by Buerschaper and Burnett [quant-ph/0406039] through numerical simulations. We also provide heuristic explanations for the behavior of our model which show, in particular, that its dynamics can be viewed as a discrete version of Bloch oscillations.
... Antoni Wójcik,1 Tomasz Łuczak,2,* Paweł Kurzyński,1 Andrzej Grudka,1 Tomasz Gdala,2 andMałgor... more ... Antoni Wójcik,1 Tomasz Łuczak,2,* Paweł Kurzyński,1 Andrzej Grudka,1 Tomasz Gdala,2 andMałgorzata Bednarska2 1Faculty of Physics, Adam Mickiewicz ... It is shown that in such a system under fairly general conditions, we can expect a nearly perfect transfer of states. ...
Let F k be the family of graphs G such that all sufficiently large k-connected claw-free graphs w... more Let F k be the family of graphs G such that all sufficiently large k-connected claw-free graphs which contain no induced copies of G are subpancyclic. We show that for every k ≥ 3 the family F 1 k is infinite and make the first step toward the complete characterization of the family F 3 . ᭧
We show that every 3-connected claw-free graph which contains no induced copy of P 11 is hamilton... more We show that every 3-connected claw-free graph which contains no induced copy of P 11 is hamiltonian. Since there exist non-hamiltonian 3-connected claw-free graphs without induced copies of P 12 this result is, in a way, best possible. ß
We characterize all pairs of connected graphs fX ; Y g such that each 3-connected fX ; Y g-free g... more We characterize all pairs of connected graphs fX ; Y g such that each 3-connected fX ; Y g-free graph is pancyclic. In particular, we show that if each of the graphs in such a pair fX ; Y g has at least four vertices, then one of them is the claw K 1;3 , while the other is a subgraph of one of six specified graphs. ß
n denote the 3-uniform tight cycle, that is the hypergraph with vertices v1, . . . , vn and edges... more n denote the 3-uniform tight cycle, that is the hypergraph with vertices v1, . . . , vn and edges v1v2v3, v2v3v4, . . . , vn−1vnv1, vnv1v2. We prove that the smallest integer N = N (n) for which every red-blue coloring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C (3) n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl.
We show that there is an absolute constant 5 > 0 such that the number of sum-free subsets of any ... more We show that there is an absolute constant 5 > 0 such that the number of sum-free subsets of any finite abelian group G is where v(G) is the number of even order components in the canonical decomposition of G into a direct sum of its cyclic subgroups, and the implicit constant in the O-sign is absolute.
Four concepts are considered for the trophic level of a species in a community food web. The long... more Four concepts are considered for the trophic level of a species in a community food web. The long-way-uplevel (or LU-level) of species A is the length of the longest simple food chain from a basal species (one with no prey in the web) to A. (A simple chain is a chain that does not pass through any given species more than once.) The short-way-up-level (SU-level) of species A is the length of the shortest chain from a basal species to A. The long-way-down-level (LD-ievel) of species A is the length of the longest simple chain from species A to a top species (one with no consumers in the web). The short-way-down-level (SD-level) of species A is the length of the shortest chain from species A to a top species. The stratigraphy of a web is the analogue for species of the 'pyramid of numbers' for individuals: it is the frequency distribution of species according to level. The LU-, SU-, LD-, and SD-stratigraphies of the seven webs in the Briand-Cohen collection with 30 or more trophic species reveal no species with LU-level or LD-level more than 6, no species with SU-level more than 3, and no species with SD-level more than 2. In all seven webs, SD-levels are stochastically less than SU-levels: species tend to be closer to a top predator than to a basal species. Two stochastic models of food web structure (the cascade model and the homogeneous superlinear model) correctly predict that 95% or more of species should have LU-level and LD-level in the range 0-6. The models also correctly predict some details of the distribution of species in the SU-and SD-stratigraphies, particularly the fraction of species in level 1. The models do not, in general, correctly predict the distribution of species within the range 0-6 of LU-levels and LD-levels.
Dobson conjectured that for every t and k a tree with k vertices can always be embedded into a gr... more Dobson conjectured that for every t and k a tree with k vertices can always be embedded into a graph of minimum degree max{ (T ); (k − 1)=t } and girth at least 2t + 1. We show that the conjecture holds for all except a ÿnite number of pairs (t; k).
Let D(G) be the smallest quantifier depth of a first-order formula which is true for a graph G bu... more Let D(G) be the smallest quantifier depth of a first-order formula which is true for a graph G but false for any other non-isomorphic graph. This can be viewed as a measure for the descriptive complexity of G in first-order logic. We show that almost surely D(G) = Θ( ln n ln ln n ), ...
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