We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by @math , which creates a shock in the particle density of order @math @math the observation time. When starting from step initial data, we provide bounds on the limiting law which in particular imply that in the double limit @math one recovers the product limit law and the degeneration of the correlation length observed at shocks of order @math . This result is shown to apply to a general last-passage percolation model. We also obtain bounds on the two-point functions of several @math processes.
A result we shall use repeatedly in this paper concerns the transversal fluctuations of the maximizer @math when @math are i.i.d. and @math there are bounds on the probability that the maximizer @math from @math to a point @math deviates more than @math from the straight line @math , @math . This is a result from @cite_11 and cited here as Theorem 3.2. Furthermore, for line-to-point problems, we have detailed control over the distribution of the (random) starting point of @math , see @cite_4 , Lemma 1.1, Lemma 1.2 and also (4.18) in @cite_1 .
@cite_1 @cite_4 @cite_11
2963279081 2952647761 117385580
We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . Given two graphs @math and @math , the Subgraph Isomorphism problem asks if @math is isomorphic to a subgraph of @math . While NP-hard in general, algorithms exist for various parameterized versions of the problem: for example, the problem can be solved (1) in time @math using the color-coding technique of Alon, Yuster, and Zwick; (2) in time @math using Courcelle's Theorem; (3) in time @math using a result on first-order model checking by Frick and Grohe; or (4) in time @math for connected @math using the algorithm of Matou s ek and Thomas. Already this small sample of results shows that the way an algorithm can depend on the parameters is highly nontrivial and subtle. We develop a framework involving 10 relevant parameters for each of @math and @math (such as treewidth, pathwidth, genus, maximum degree, number of vertices, number of components, etc.), and ask if an algorithm with running time [ f_1(p_1,p_2,..., p_ ) n^ f_2(p_ +1 ,..., p_k) ] exist, where each of @math is one of the 10 parameters depending only on @math or @math . We show that all the questions arising in this framework are answered by a set of 11 maximal positive results (algorithms) and a set of 17 maximal negative results (hardness proofs); some of these results already appear in the literature, while others are new in this paper. On the algorithmic side, our study reveals for example that an unexpected combination of bounded degree, genus, and feedback vertex set number of @math gives rise to a highly nontrivial algorithm for Subgraph Isomorphism. On the hardness side, we present W[1]-hardness proofs under extremely restricted conditions, such as when @math is a bounded-degree tree of constant pathwidth and @math is a planar graph of bounded pathwidth. The celebrated Time Hierarchy Theorem for Turing machines states, informally, that more problems can be solved given more time. The extent to which a time hierarchy-type theorem holds in the distributed LOCAL model has been open for many years. It is consistent with previous results that all natural problems in the LOCAL model can be classified according to a small constant number of complexities, such as @math , etc. In this paper we establish the first time hierarchy theorem for the LOCAL model and prove that several gaps exist in the LOCAL time hierarchy. 1. We define an infinite set of simple coloring problems called Hierarchical @math -Coloring . A correctly colored graph can be confirmed by simply checking the neighborhood of each vertex, so this problem fits into the class of locally checkable labeling (LCL) problems. However, the complexity of the @math -level Hierarchical @math -Coloring problem is @math , for @math . The upper and lower bounds hold for both general graphs and trees, and for both randomized and deterministic algorithms. 2. Consider any LCL problem on bounded degree trees. We prove an automatic-speedup theorem that states that any randomized @math -time algorithm solving the LCL can be transformed into a deterministic @math -time algorithm. Together with a previous result, this establishes that on trees, there are no natural deterministic complexities in the ranges @math --- @math or @math --- @math . 3. We expose a gap in the randomized time hierarchy on general graphs. Any randomized algorithm that solves an LCL problem in sublogarithmic time can be sped up to run in @math time, which is the complexity of the distributed Lovasz local lemma problem, currently known to be @math and @math . We study competing first passage percolation on graphs generated by the configuration model. At time 0, vertex 1 and vertex 2 are infected with the type 1 and the type 2 infection, respectively, and an uninfected vertex then becomes type 1 (2) infected at rate @math ( @math ) times the number of edges connecting it to a type 1 (2) infected neighbor. Our main result is that, if the degree distribution is a power-law with exponent @math , then, as the number of vertices tends to infinity and with high probability, one of the infection types will occupy all but a finite number of vertices. Furthermore, which one of the infections wins is random and both infections have a positive probability of winning regardless of the values of @math and @math . The picture is similar with multiple starting points for the infections. When graph Ramsey theory is viewed as a game, "Painter" 2-colors the edges of a graph presented by "Builder". Builder wins if every coloring has a monochromatic copy of a fixed graph @math . In the on-line version, iteratively, Builder presents one edge and Painter must color it. Builder must keep the presented graph in a class @math . Builder wins the game @math if a monochromatic copy of @math can be forced. The on-line degree Ramsey number @math is the least @math such that Builder wins @math when @math is the class of graphs with maximum degree at most @math . Our results include: 1) @math if and only if @math is a linear forest or each component lies inside @math . 2) @math , where @math . 3) @math for a tree @math , where @math and @math are two largest vertex degrees. 4) @math , with @math except for finitely many odd values of @math . 5) @math when @math . The lower bounds come from strategies for Painter that color edges red whenever the red graph remains in a specified class. The upper bounds use a result showing that Builder may assume that Painter plays "consistently". We derive a new upper bound on the diameter of a polyhedron @math P = x ? R n : A x ≤ b , where @math A ? Z m × n . The bound is polynomial in @math n and the largest absolute value of a sub-determinant of @math A , denoted by @math Δ . More precisely, we show that the diameter of @math P is bounded by @math O ( Δ 2 n 4 log n Δ ) . If @math P is bounded, then we show that the diameter of @math P is at most @math O ( Δ 2 n 3.5 log n Δ ) . For the special case in which @math A is a totally unimodular matrix, the bounds are @math O ( n 4 log n ) and @math O ( n 3.5 log n ) respectively. This improves over the previous best bound of @math O ( m 16 n 3 ( log m n ) 3 ) due to Dyer and Frieze (Math Program 64:1---16, 1994). A perfect straight-line matching @math on a finite set @math of points in the plane is a set of segments such that each point in @math is an endpoint of exactly one segment. @math is non-crossing if no two segments in @math cross each other. Given a perfect straight-line matching @math with at least one crossing, we can remove this crossing by a flip operation. The flip operation removes two crossing segments on a point set @math and adds two non-crossing segments to attain a new perfect matching @math . It is well known that after a finite number of flips, a non-crossing matching is attained and no further flip is possible. However, prior to this work, no non-trivial upper bound on the number of flips was known. If @math (resp. @math ) is the maximum length of the longest (resp. shortest) sequence of flips starting from any matching of size @math , we show that @math and @math (resp. @math and @math ). We prove an incidence theorem for points and planes in the projective space @math over any field @math , whose characteristic @math An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving @math if @math . This yields a bound on the number of incidences between @math points and @math planes in @math , with @math as @math where @math is the maximum number of collinear planes, provided that @math if @math . Examples show that this bound cannot be improved without additional assumptions. This gives one a vehicle to establish geometric incidence estimates when @math . For a non-collinear point set @math and a non-degenerate symmetric or skew-symmetric bilinear form @math , the number of distinct values of @math on pairs of points of @math is @math . This is also the best known bound over @math , where it follows from the Szemer 'edi-Trotter theorem. Also, a set @math , not supported in a single semi-isotropic plane contains a point, from which @math distinct distances to other points of @math are attained. The @math -online Ramsey game is a combinatorial game between two players, Builder and Painter. Starting from an infinite set of isolated vertices, Builder draws an edge at each step and Painter immediately paints the edge red or blue. Builder's goal is to force Painter to create either a red @math or a blue @math using as few edges as possible. The online Ramsey number @math is the minimum number of edges Builder needs to guarantee a win in the @math -online Ramsey game. By analyzing the special case where Painter plays randomly, we obtain an exponential improvement [ r (n,n) 2^ (2- 2 )n + O(1) ] for the lower bound on the diagonal online Ramsey number, as well as a corresponding improvement [ r (m,n) n^ (2- 2 )m + O(1) ] for the off-diagonal case, where @math is fixed and @math . Using a different randomized Painter strategy and the Lopsided Lov 'asz Local Lemma, we prove that @math , determining this function up to a polylogarithmic factor. We also improve the upper bound in the off-diagonal case for @math . In connection with the online Ramsey game with random Painter, we study the problem of finding a copy of a target graph @math in a sufficiently large unknown Erd o s--R ' e nyi random graph @math using as few queries as possible, where each query reveals whether or not a particular pair of vertices are adjacent. We call this problem the Subgraph Query Problem. We determine the order of the number of queries needed for complete graphs up to five vertices and prove general bounds for this problem. Introduction Preliminaries Special linear groups: @math Orthogonal groups: @math Symplectic groups: @math Unitary groups: @math Proofs of Theorems 1.1 and 1.1, and of corollaries 1.2-1.4 Permutation group algorithms Concluding remarks References. What is the densest packing of points in an infinite strip of widthw, where any two of the points must be separated by distance at least I? This question was raised by Fejes-Toth a number of years ago. The answer is trivial for MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa83Dai abgsMiJoaakaaabaGaaG4maaWcbeaakiaac+cacaaIYaaaaa!3AF6! @math and, surprisingly, it is not difficult to prove [M2] for MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa83Dai aa-1dacaWGUbWaaOaaaeaacaaIZaaaleqaaOGaai4laiaaikdaaaa!3AF2! @math , wheren is a positive integer, that the regular triangular lattice gives the optimal packing. Kertesz [K] solved the case MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa83DaG qaaiaa+bcacaGF8aWaaOaaaeaacaaIYaaaleqaaaaa!392C! @math . Here we fill the first gap, i.e., the maximal density is determined for MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaie aacaWFZaaaleqaaOGaai4laiaaikdacqGH8aapcaWG3bGaeyizIm6a aOaaaeaacaaIZaaaleqaaaaa!3CCB! @math . Let @math be independent random positive variables and let @math , @math Let us denote [ P _1 + + _n < t = G_n (t). ]Theorem. [ t 1 - G_n (t) 1 - G(t) = n, n = 1,2,3, , ]if and only if [ t 1 - G_2 (t) 1 - G(t) = 2. ] This theorem is useful in some investigations of age-dependent branching processes. In this paper, we study online algorithms that schedule malleable jobs, i.e., jobs that can be parallelized on any subset of the available @math m identical machines. We study a model that accounts for the tradeoff between multiprocessor speedup and overhead time, namely, if job @math j has processing requirement @math pj and is assigned to run on @math kj machines, then @math j's execution time becomes @math pj kj+(kj-1)c, where @math c is a constant parameter to the problem. For @math m=2, we present an online algorithm OCS that has a strong competitive ratio of 3 2, matching a previously established lower bound. We also present an online algorithm ASYM2 that is asymptotically @math ((4-∈) (3-∈))-competitive when @math m=2, where @math 0<∈≤2 is a parameter to the algorithm, improving upon an existing asymptotically (3 2)-competitive algorithm. Finally, we present an online algorithm OTO that is strongly @math 2-competitive when @math m=3, improving upon the previous best upper bound of @math 9 4. Let @math be an i.i.d. sequence of random variables in @math , @math . We show that, for any function @math , under regularity conditions, [n^ 1 2 (n^ -1 i=1 ^n (X_i) f ^(X_i) - (x) ,dx ) P 0, ] where @math is the classical kernel estimator of the density of @math . This result is striking because it speeds up traditional rates, in root @math , derived from the central limit theorem when @math . Although this paper highlights some applications, we mainly address theoretical issues related to the later result. We derive upper bounds for the rate of convergence in probability. These bounds depend on the regularity of the functions @math and @math , the dimension @math and the bandwidth of the kernel estimator @math . Moreover, they are shown to be accurate since they are used as renormalizing sequences in two central limit theorems each reflecting different degrees of smoothness of @math . As an application to regression modelling with random design, we provide the asymptotic normality of the estimation of the linear functionals of a regression function. As a consequence of the above result, the asymptotic variance does not depend on the regression function. Finally, we debate the choice of the bandwidth for integral approximation and we highlight the good behavior of our procedure through simulations. The ( @math 1+ź) EA with mutation probability c n, where @math c>0 is an arbitrary constant, is studied for the classical OneMax function. Its expected optimization time is analyzed exactly (up to lower order terms) as a function of c and @math ź. It turns out that 1 n is the only optimal mutation probability if @math ź=o(lnnlnlnn lnlnlnn), which is the cut-off point for linear speed-up. However, if @math ź is above this cut-off point then the standard mutation probability 1 n is no longer the only optimal choice. Instead, the expected number of generations is (up to lower order terms) independent of c, irrespectively of it being less than 1 or greater. The theoretical results are obtained by a careful study of order statistics of the binomial distribution and variable drift theorems for upper and lower bounds. Experimental supplements shed light on the optimal mutation probability for small problem sizes. In this paper we study general @math l p regularized unconstrained minimization problems. In particular, we derive lower bounds for nonzero entries of the first- and second-order stationary points and hence also of local minimizers of the @math l p minimization problems. We extend some existing iterative reweighted @math l 1 ( @math IRL 1 ) and @math l 2 ( @math IRL 2 ) minimization methods to solve these problems and propose new variants for them in which each subproblem has a closed-form solution. Also, we provide a unified convergence analysis for these methods. In addition, we propose a novel Lipschitz continuous @math ∈ -approximation to @math ? x ? p p . Using this result, we develop new @math IRL 1 methods for the @math l p minimization problems and show that any accumulation point of the sequence generated by these methods is a first-order stationary point, provided that the approximation parameter @math ∈ is below a computable threshold value. This is a remarkable result since all existing iterative reweighted minimization methods require that @math ∈ be dynamically updated and approach zero. Our computational results demonstrate that the new @math IRL 1 method and the new variants generally outperform the existing @math IRL 1 methods (Chen and Zhou in 2012; Foucart and Lai in Appl Comput Harmon Anal 26:395---407, 2009). Deep neural networks (DNNs) have begun to have a pervasive impact on various applications of machine learning. However, the problem of finding an optimal DNN architecture for large applications is challenging. Common approaches go for deeper and larger DNN architectures but may incur substantial redundancy. To address these problems, we introduce a network growth algorithm that complements network pruning to learn both weights and compact DNN architectures during training. We propose a DNN synthesis tool (NeST) that combines both methods to automate the generation of compact and accurate DNNs. NeST starts with a randomly initialized sparse network called the seed architecture. It iteratively tunes the architecture with gradient-based growth and magnitude-based pruning of neurons and connections. Our experimental results show that NeST yields accurate, yet very compact DNNs, with a wide range of seed architecture selection. For the LeNet-300-100 (LeNet-5) architecture, we reduce network parameters by @math 70.2× ( @math 74.3×) and floating-point operations (FLOPs) by @math 79.4× ( @math 43.7×). For the AlexNet, VGG-16, and ResNet-50 architectures, we reduce network parameters (FLOPs) by @math 15.7× ( @math 4.6×), @math 33.2× ( @math 8.9×), and @math 4.1× ( @math 2.1×) respectively. NeST's grow-and-prune paradigm delivers significant additional parameter and FLOPs reduction relative to pruning-only methods. In this paper, we propose a framework to study a general class of strategic behavior in voting, which we call vote operations. We prove the following theorem: if we fix the number of alternatives, generate @math votes i.i.d. according to a distribution @math , and let @math go to infinity, then for any @math , with probability at least @math , the minimum number of operations that are needed for the strategic individual to achieve her goal falls into one of the following four categories: (1) 0, (2) @math , (3) @math , and (4) @math . This theorem holds for any set of vote operations, any individual vote distribution @math , and any integer generalized scoring rule, which includes (but is not limited to) almost all commonly studied voting rules, e.g., approval voting, all positional scoring rules (including Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs. We also show that many well-studied types of strategic behavior fall under our framework, including (but not limited to) constructive destructive manipulation, bribery, and control by adding deleting votes, margin of victory, and minimum manipulation coalition size. Therefore, our main theorem naturally applies to these problems. Let @math be a compact Riemannian submanifold of @math of dimension @math and let @math be a sample of i.i.d. points in @math with uniform distribution. We study the random operators @math where @math is the Gaussian kernel and @math as @math Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approximate the Laplace-Beltrami operator of @math @math (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations @math for smooth functions @math both pointwise and uniformly in @math and @math (here @math and @math is the Riemannian volume measure). In particular, we show that for any class @math of three times differentiable functions on @math with uniformly bounded derivatives @math as soon as @math and also prove asymptotic normality of @math (functional CLT) for a fixed @math and uniformly in @math We address the problem of replicating a Voronoi diagram @math of a planar point set @math by making proximity queries, which are of three possible (in decreasing order of information content): 1. the exact location of the nearest site(s) in @math ; 2. the distance to and label(s) of the nearest site(s) in @math ; 3. a unique label for every nearest site in @math . We provide algorithms showing how queries of Type 1 and Type 2 allow an exact cloning of @math with @math queries and @math processing time. We also prove that queries of Type 3 can never exactly clone @math , but we show that with @math queries we can construct an @math -approximate cloning of @math . In addition to showing the limits of nearest-neighbor database security, our methods also provide one of the first natural algorithmic applications of retroactive data structures.