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Nowadays, pictures have more and more applications in various fields. Because different situations may have different commands of the ratios of pictures, we present a program which can resize the pictures according to their contents. Compared with the traditional method which only cares about the geometric constraints, our program can reserve the important contents of the pictures, which means that the visual effect of the pictures will not be damaged after the resizing. In order to change the ratio of pictures while avoiding distorting their main contents, we use a method called seam carving. Based on the energy function used for finding useless pixels and dynamic programming used for finding some continuous seams which will be deleted later, our program can finally change the image to a new ratio perfectly. Due to some deficiencies of the basic program, we also make some improvements. The first one is the improvement of the algorithm: we add face detection to the basic program. Not only does this improvement can solve many problems of the basic program automatically, but also it is our own innovation of the study of content-aware resizing. The second one is the addition of customization: we allow users to select the specific contents that they want to reserve, which can meet the different needs of users. It’s also our own innovation which will be very useful in practical situations. Our program can have many practical applications. We have some ideas about how to put this algorithm into real use, which is one of our practical innovations of this topic. In terms of different ratios of photo printing, graphic designing and screen displaying, this program will have various uses, which will make a groundbreaking progress in practice.

A classification of $\SLn$ contravariant, continuous function-valued valuations on convex bodies is established. Such valuations are natural extensions of $\SLn$ contravariant $L_p$ Minkowski valuations, the classification of which characterized $L_p$ projection bodies, which are fundamental in the $L_p$ Brunn-Minkowski theory, for $p \geq 1$. Hence our result will help to better understand extensions of the $L_p$ Brunn-Minkowski theory. In fact, our results characterize general projection functions which extend $L_p$ projection functions ($p$-th powers of the support functions of $L_p$ projection bodies) to projection functions in the $L_p$ Brunn-Minkowski theory for $0< p < 1$ and in the Orlicz Brunn-Minkowski theory.

In this paper, we consider the discontinuous Galerkin (DG) methods to solve linear hyperbolic equations with singular initial data. With the help of weight functions, the superconvergence properties outside the pollution region will be investigated. We show that, by using piecewise polynomials of degree $k$ and suitable initial discretizations, the DG solution is $(2k+1)$-th order accurate at the downwind points and $(k+2)$-th order accurate at all the other downwind-biased Radau points. Moreover, the derivative of error between the DG and exact solutions converges at a rate of $k+1$ at all the interior upwind-biased Radau points. Besides the above, the DG solution is also $(k+2)$-th order accurate towards a particular projection of the exact solution and the numerical cell averages are $(2k+1)$-th order accurate. Numerical experiments are presented to confirm the theoretical results.

We introduce a new technique for isolating components on the spectral side of the trace formula. By applying it to the Jacquet--Rallis relative trace formula, we complete the proof of the global Gan--Gross--Prasad conjecture and its refinement Ichino--Ikeda conjecture for U(n)×U(n+1) in the stablecase.

In this paper we explicitly construct Moishezon twistor spaces on nCP2 for arbitrary n ≥ 2 which admit a holomorphic Caction. When n = 2, they coincide with Y. Poon’s twistor spaces. When n = 3, they coincide with the ones studied by the author in [14]. When n ≥ 4, they are new twistor spaces, to the best of the author’s knowledge. By investigating the anticanonical system, we show that our twistor spaces are bimeromorphic to conic bundles over certain rational surfaces. The latter surfaces can be regarded as orbit spaces of the C-action on the twistor spaces. Namely they are minitwistor spaces. We explicitly determine their defining equations in CP4. It turns out that the structure of the minitwistor space is independent of n. Further, we explicitly construct a CP2-bundle over the resolution of this surface, and provide an explicit defining equation of the conic bundles. It shows that the number of irreducible components of the discriminant locus for the conic bundles increases as n does. Thus our twistor spaces have a lot of similarities with the famous LeBrun twistor spaces, where the minitwistor space CP1 ×CP1 in LeBrun’s case is replaced by our minitwistor spaces found in [15].

For a one-parameter family (V, {i}pg i=1) of general type hypersurfaces with bases of holomorphic n-forms, we construct open covers V = Spg i=1 Ui using tropical geometry. We show that after normalization, each i is approximately supported on a unique Ui and such a pair approximates a Calabi-Yau hypersurface together with its holomorphic n-form as the parameter becomes large. We also show that the Lagrangian fibers in the fibration constructed by Mikhalkin [9] are asymptotically special Lagrangian. As the holomorphic n-form plays an important role in mirror symmetry for Calabi-Yau manifolds, our results is a step toward understanding mirror symmetry for general type manifolds.

No abstract uploaded!

No abstract uploaded!

The operators with large scaling dimensions can be labeled by Young diagrams. Among other bases, the operators using restricted Schur polynomials have been known to have a large N but nonplanar limit under which they map to states of a system of harmonic oscillators. We analyze the oscillator algebra acting on pairs of long rows or long columns in the Young diagrams of the operators. The oscillator algebra can be reached by a Inonu– Wigner contraction of the u(2) algebra inside of the u(p) algebra of p giant gravitons. We present evidences that integrability in this case can persist at higher loops due to the presence of the oscillator algebra which is expected to be robust under loop corrections in the nonplanar large N limit.

In this paper, we study some relation between the cocenter H(G) of the Hecke algebra H (G) of a connected reductive group G over a nonarchimedean local field and the cocenter H(M) of its Levi subgroups M. Given any Newton component of H(G), we construct the induction map i from the corresponding Newton component of H(M) to it. We show that this map is an isomorphism. This leads to the BernsteinLusztig type presentation of the cocenter H(G), which generalizes the work [11] on the affine Hecke algebras. We also show that the map i we constructed is adjoint to the Jacquet functor.

Varying coefficient models are a useful extension of classical linear models. They arise naturally when one wishes to examine how regression coefficients change over different groups characterized by certain covariates such as age. The appeal of these models is that the coef. cient functions can easily be estimated via a simple local regression. This yields a simple one-step estimation procedure. We show that such a one-step method cannot be optimal when different coefficient functions admit different degrees of smoothness. This drawback can be repaired by using our proposed two-step estimation procedure. The asymptotic mean-squared error for the two-step procedure is obtained and is shown to achieve the optimal rate of convergence. A few simulation studies show that the gain by the two-step procedure can be quite substantial. The methodology is illustrated by an application to an environmental data set.

Measuring conditional dependence is an important topic in statistics with broad applications including graphical models. Under a factor model setting, a new conditional dependence measure based on projection is proposed. The corresponding conditional independence test is developed with the asymptotic null distribution unveiled where the number of factors could be high-dimensional. It is also shown that the new test has control over the asymptotic significance level and can be calculated efficiently. A generic method for building dependency graphs without Gaussian assumption using the new test is elaborated. Numerical results and real data analysis show the superiority of the new method.

Let <i>L</i> be a holomorphic line bundle over a compact Khler manifold <i>X</i>. Motivated by mirror symmetry, we study the deformed HermitianYangMills equation on <i>L</i>, which is the line bundle analogue of the special Lagrangian equation in the case that <i>X</i> is CalabiYau. We show that this equation is the Euler-Lagrange equation for a positive functional, and that solutions are unique global minimizers. We provide a necessary and sufficient criterion for existence in the case that <i>X</i> is a Khler surface. For the higher dimensional cases, we introduce a line bundle version of the Lagrangian mean curvature flow, and prove convergence when <i>L</i> is ample and <i>X</i> has non-negative orthogonal bisectional curvature.

We study the two-point correlation function of a freely decaying scalar in Kraichnan's model of advection by a Gaussian random velocity field that is stationary and white noise in time, but fractional Brownian in space with roughness exponent 0&lt;<i></i>&lt;2, appropriate to the inertial-convective range of the scalar. We find all self-similar solutions by transforming the scaling equation to Kummer's equation. It is shown that only those scaling solutions with scalar energy decay exponent <i>a</i>(<i>d</i>/<i></i>)+1 are statistically realizable, where <i>d</i> is space dimension and <i></i>=2<i></i>. An infinite sequence of invariants <i>J</i> _p, <i>p</i>=0,1,2,..., is pointed out, where <i>J</i> ₀ is Corrsin's integral invariant but the higher invariants appear to be new. We show that at least one of the invariants <i>J</i> ₀ or <i>J</i> ₁ must be nonzero (possibly infinite) for realizable initial data. Initial datum with a finite, nonzero invariantthe first being <i>J</i> _pconverges at long times to a scaling

We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in temporally random shear flows with sufficiently decaying correlations. A key quantity in the variational principle is the almost sure Lyapunov exponent of a heat operator with random potential. To prove the variational principle, we use the comparison principle of solutions, the path integral representation of solutions, and large deviation estimates of the associated stochastic flows. The variational principle then allows us to analytically bound the front speeds. The speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length, and the sublinear growth law if the temporal decorrelation is also large enough, the so-called bending phenomenon.

We prove that classical solution of the spatially inhomogeneous and angular non-cutoff Boltzmann equation is C with respect to all variables, locally in the space and time variables. The proof relies on a generalized uncertainty principle, some improved upper bound and coercivity estimates on the nonlinear collision operator, and some subtle analysis on the commutators between the collision operators and some appropriately chosen pseudo-differential operators. To cite this article: R. Alexandre et al., CR Acad. Sci. Paris, Ser. I 347 (2009).

We develop an auto-reservoir computing framework, Auto-Reservoir Neural Network (ARNN), to efficiently and accurately make multi-step-ahead predictions based on a short-term high-dimensional time series. Different from traditional reservoir computing whose reservoir is an external dynamical system irrelevant to the target system, ARNN directly transforms the observed high-dimensional dynamics as its reservoir, which maps the high-dimensional/spatial data to the future temporal values of a target variable based on our spatiotemporal information (STI) transformation. Thus, the multi-step prediction of the target variable is achieved in an accurate and computationally efficient manner. ARNN is successfully applied to both representative models and real-world datasets, all of which show satisfactory performance in the multi-step-ahead prediction, even when the data are peturbed by noise and when the system is time-varying. Actually, such ARNN transformation equivalently expands the sample size and thus has great potential in practical applications in artificial intelligence and machine learning.

In this article, we define a new class of middle dimensional submanifolds of a Hyperkahler manifold which contains the class of complex Lagrangian submanifolds, and show that this larger class is invariant under the mean curvature flow. Along the flow, the complex phase map satisfies the generalized harmonic map heat equation, tt is also related to the mean curvature vector via a first order differential equation. Moreover, we proved a result on nonexistence of Type I singularity.

We prove a finiteness property of the values of the skein polynomial of homogeneous knots that allows us to establish large classes of such knots to have arbitrarily unsharp Bennequin inequality (for the Thurston-Bennequin invariant of any of their Legendrian embeddings in the standard contact structure of R3). We also give a short proof that there are only finitely many such knots that have given genus and given braid index.

In this paper we study constant mean curvature surfaces  in a product space, M2 × R, where M2 is a complete Riemannian manifold. We assume the angle function ν = hN, @ @t i does not change sign on . We classify these surfaces according to the infimum c() of the Gaussian curvature of the projection of . When H 6= 0 and c() ≥ 0, then  is a cylinder over a complete curve with curvature 2H. If H = 0 and c() ≥ 0, then  must be a vertical plane or  is a slice M2 ×{t}, or M2 ≡ R2 with the flat metric and  is a tilted plane (after possibly passing to a covering space). When c() < 0 and H > p −c()/2, then  is a vertical cylinder over a complete curve of M2 of constant geodesic curvature 2H. This result is optimal. We also prove a non-existence result concerning complete multigraphs in M2 × R, when c(M2) < 0.

No abstract uploaded!

No abstract uploaded!

In this paper, we study the relation between topological orders and their gapped boundaries. We propose that the bulk for a given gapped boundary theory is unique. It is actually a consequence of a microscopic definition of a local topological order, which is a (potentially anomalous) topological order defined on an open disk. Using this uniqueness, we show that the notion of “bulk” is equivalent to the notion of center in mathematics. We achieve this by first introducing the notion of a morphism between two local topological orders of the same dimension, then proving that the bulk satisfying the same universal property as that of the center in mathematics. We propose a classification (formulated as a macroscopic definition of n+1D local topological orders by unitary multi-fusion n-categories, and explain that the notion of a morphism between two local topological orders is compatible with that of a unitary monoidal n-functor in a few low dimensional cases. We also explain in some low dimensional cases that this classification is compatible with the result of “bulk = center”. In the end, we explain that above boundary-bulk relation is only the first layer of a hierarchical structure which can be summarized by the functoriality of the bulk (or center). This functoriality also provides the physical meanings of some well-known mathematical results on fusion 1-categories. This work can also be viewed as the first step towards a systematic study of the category of local topological orders, and the boundary-bulk relation actually provides a useful tool for this study.

Personal videos often contain visual distractors, which are objects that are accidentally captured that can distract viewers from focusing on the main subjects. We propose a method to automatically detect and localize these distractors through learning from a manually labeled dataset. To achieve spatially and temporally coherent detection, we propose extracting features at the Temporal-Superpixel (TSP) level using a traditional SVM-based learning framework. We also experiment with end-to-end learning using Convolutional Neural Networks (CNNs), which achieves slightly higher performance than other methods. The classification result is further refined in a post-processing step based on graph-cut optimization. Experimental results show that our method achieves an accuracy of 81% and a recall of 86%. We demonstrate several ways of removing the detected distractors to improve the video quality, including video hole filling; video frame replacement; and camera path re-planning. The user study results show that our method can significantly improve the aesthetic quality of videos.