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Turán's theorem is a cornerstone of extremal graph theory. It asserts that for any integer r ≥ 2, every graph on n vertices with more than $\\frac{{r - 2}}{{2\\left( {r - 1} \\right)}} \\cdot {n^2}$ edges contains a clique of size r, i.e., r mutually adjacent vertices. The corresponding extremal graphs are balanced (r – 1)-Partite graphs. The question as to how many such r-cliques appear at least in any n-vertex graph with γn² edges has been intensively studied in the literature. In particular, Lovász and Simonovits conjectured in the 1970's that asymptotically the best possible lower bound is given by the complete multipartite graph with γn² edges in which all but one vertex class is of the same size while the remaining one may be smaller. Their conjecture was recently resolved for r = 3 by Razborov and for r = 4 by Nikiforov. In this article, we prove the conjecture for all values of r.

Let 0 < 𝑠 < ∞. In this study, we introduce the double sequence space 𝑅𝑞𝑡(𝓛𝑠) as the domain of four dimensional Riesz mean 𝑅𝑞𝑡in the space 𝓛𝑠of absolutely 𝑠-summable double sequences. Furthermore, we show that 𝑅𝑞𝑡(𝓛𝑠) is a Banach space and a barrelled space for 1 ≤ 𝑠 < ∞ and is not a barrelled space for 0 < 𝑠 < 1. We determine the 𝛼- and 𝛽(𝑣)-duals of the space 𝓛𝑠for 0 < 𝑠 ≤ 1 and 𝛽(𝑏𝑝)-dual of the space 𝑅𝑞𝑡(𝓛𝑠) for 1 < 𝑠 < ∞, where 𝑣∈{𝑝,𝑏𝑝,𝑟}. Finally, we characterize the classes (𝓛𝑠: 𝛭𝑢), (𝓛𝑠: 𝐶𝑏𝑝), (𝑅𝑞𝑡(𝓛𝑠) : 𝛭𝑢) and (𝑅𝑞𝑡(𝓛𝑠) : 𝐶𝑏𝑝) of four dimensional matrices in the cases both 0 < 𝑠 < 1 and 1 ≤ 𝑠 < ∞ together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

We prove a bubble-neck decomposition together with an energy quantization result for sequences of Willmore surfaces into ℝm with uniformly bounded energy and nondegenerating conformal type. We deduce the strong compactness of Willmore closed surfaces of a given genus modulo the Möbius group action, below some energy threshold.

We show that multiple polynomial ergodic averages arising from nilpotent groups of measure preserving transformations of a probability space always converge in the L 2 norm.

We develop a framework in which Szemerédi’s celebrated Regularity Lemma for graphs interacts with core model-theoretic ideas and techniques. Our work relies on a coincidence of ideas from model theory and graph theory: arbitrarily large half-graphs coincide with model-theoretic instability, so in their absence, structure theorems and technology from stability theory apply. In one direction, we address a problem from the classical Szemerédi theory. It was known that the “irregular pairs” in the statement of Szemerédi’s Regularity Lemma cannot be eliminated, due to the counterexample of half-graphs (i.e., the order property, corresponding to model-theoretic instability). We show that half-graphs are the only essential difficulty, by giving a much stronger version of Szemerédi’s Regularity Lemma for models of stable theories of graphs (i.e. graphs with the non-k∗k_*-order property), in which there are no irregular pairs, the bounds are significantly improved, and each component satisfies an indivisibility condition. In the other direction, we take a more model-theoretic approach, and give several new Szemerédi-type partition theorems for models of stable theories of graphs. The first theorem gives a partition of any such graph into indiscernible components, meaning here that each component is either a complete or an empty graph, whose interaction is strongly uniform. This relies on a finitary version of the classic model-theoretic fact that stable theories admit large sets of indiscernibles, by showing that in models of stable theories of graphs one can extract much larger indiscernible sets than expected by Ramsey’s theorem. The second and third theorems allow for a much smaller number of components at the cost of weakening the “indivisibility” condition on the components. We also discuss some extensions to graphs without the independence property. All graphs are finite and all partitions are equitable, i.e. the sizes of the components differ by at most 1. In the last three theorems, the number of components depends on the size of the graph; in the first theorem quoted, this number is a function of ϵ\\epsilon only as in the usual Szemerédi Regularity Lemma.

We introduce a new iterative algorithm to find sparse solutions of underdetermined linear systems. The algorithm, a simple combination of the Iterative Hard Thresholding algorithm and the Compressive Sampling Matching Pursuit algorithm, is called Hard Thresholding Pursuit. We study its general convergence and notice in particular that only a finite number of iterations are required. We then show that, under a certain condition on the restricted isometry constant of the matrix of the linear system, the Hard Thresholding Pursuit algorithm indeed finds all s-sparse solutions. This condition, which reads ${\\delta _{{3_3}}}{\\rm{ }}1/\\sqrt {3,} $ is heuristically better than the sufficient conditions currently available for other compressive sensing algorithms. It applies to fast versions of the algorithm, too, including the Iterative Hard Thresholding algorithm. Stability with respect to sparsity defect and robustness with respect to measurement error are also guaranteed under the condition ${\\delta _{{3_3}}} < 1/\\sqrt {3,} $ . We conclude with some numerical experiments to demonstrate the good empirical performance and the low complexity of the Hard Thresholding Pursuit algorithm.

We begin by studying inventory accumulation at a LIFO (last-in-first-out) retailer with two products. In the simplest version, the following occur with equal probability at each time step: first product ordered, first product produced, second product ordered, second product produced. The inventory thus evolves as a simple random walk on ℤ². In more interesting versions, a p fraction of customers orders the \"freshest available\" product regardless of type. We show that the corresponding random walks scale to Brownian motions with diffusion matrices depending on p. We then turn our attention to the critical Fortuin-Kastelyn random planar map model, which gives, for each q > 0, a probability measure on random (discretized) two-dimensional surfaces decorated by loops, related to the q-state Potts model. A longstanding open problem is to show that as the discretization gets finer, the surfaces converge in law to a limiting (loop-decorated) random surface. The limit is expected to be a Liouville quantum gravity surface decorated by a conformai loop ensemble, with parameters depending on q. Thanks to a bijection between decorated planar maps and inventory trajectories (closely related to bijections of Bernardi and Mullin), our results about the latter imply convergence of the former in a particular topology. A phase transition occurs at p = 1/2, q = 4.

In this paper, we define the superposition operator 𝑃𝑔where 𝑔 : ℕ² × ℝ → ℝ by 𝑃𝑔((𝑥𝑘𝑠)) = 𝑔 (𝑘,𝑠,𝑥𝑘𝑠) for all real double sequence (𝑥𝑘𝑠). Chew & Lee [4] and Petranuarat & Kemprasit [7] have characterized 𝑃𝑔: 𝑙𝑝→ 𝑙₁ and 𝑃𝑔: 𝑙𝑝→ 𝑙𝑞where 1 ≤ 𝑝,𝑞 < ∞, respectively. The main goal of this paper is to construct the necessary and sufficient conditions for the continuity of 𝑃𝑔: ℒ𝑝→ ℒ₁ and 𝑃𝑔: ℒ𝑝→ ℒ𝑞where 1 ≤ 𝑝,𝑞 < ∞.

A longstanding debate concerns the use of concrete versus abstract instructional materials, particularly in domains such as mathematics and science. Although decades of research have focused on the advantages and disadvantages of concrete and abstract materials considered independently, we argue for an approach that moves beyond this dichotomy and combines their advantages. Specifically, we recommend beginning with concrete materials and then explicitly and gradually fading to the more abstract. Theoretical benefits of this \"concreteness fading\" technique for mathematics and science instruction include (1) helping learners interpret ambiguous or opaque abstract symbols in terms of well-understood concrete objects, (2) providing embodied perceptual and physical experiences that can ground abstract thinking, (3) enabling learners to build up a store of memorable images that can be used when abstract symbols lose meaning, and (4) guiding learners to strip away extraneous concrete properties and distill the generic, generalizable properties. In these ways, concreteness fading provides advantages that go beyond the sum of the benefits of concrete and abstract materials.

Bacteria thrive on and within the human body. One of the largest human-associated microbial habitats is the skin surface, which harbors large numbers of bacteria that can have important effects on health. We examined the palmar surfaces of the dominant and nondominant hands of 51 healthy young adult volunteers to characterize bacterial diversity on hands and to assess its variability within and between individuals. We used a novel pyrosequencing-based method that allowed us to survey hand surface bacterial communities at an unprecedented level of detail. The diversity of skin-associated bacterial communities was surprisingly high; a typical hand surface harbored >150 unique species-level bacterial phylotypes, and we identified a total of 4,742 unique phylotypes across all of the hands examined. Although there was a core set of bacterial taxa commonly found on the palm surface, we observed pronounced intra- and interpersonal variation in bacterial community composition: hands from the same individual shared only 17% of their phylotypes, with different individuals sharing only 13%. Women had significantly higher diversity than men, and community composition was significantly affected by handedness, time since last hand washing, and an individual's sex. The variation within and between individuals in microbial ecology illustrated by this study emphasizes the challenges inherent in defining what constitutes a \"healthy\" bacterial community; addressing these challenges will be critical for the International Human Microbiome Project.