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We define rigorously a solution to the fourth-order total variation flow equation in R n . If n ≥ 3 , it can be understood as a gradient flow of the total variation energy in D − 1 , the dual space of D 0 1 , which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case n ≤ 2 , the space D − 1 does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if n ≠ 2. If n ≠ 2 , all annuli are calibrable, while in the case n = 2 , if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs. We define rigorously a solution to the fourth-order total variation flow equation in . If , it can be understood as a gradient flow of the total variation energy in , the dual space of , which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case , the space does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if . If , all annuli are calibrable, while in the case , if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs. [ABSTRACT FROM AUTHOR]

We consider weakly and strongly asymptotically mean value harmonic (amv-harmonic) functions on subriemannian and RCD settings. We demonstrate that, in non-collapsed RCD-spaces with vanishing metric measure boundary, Cheeger harmonic functions are weakly amv-harmonic and that, in Carnot groups, weak amv-harmonicity equivalently characterizes harmonicity in the sense of the sub-Laplacian. In homogeneous Carnot groups of step 2, we prove a Blaschke–Privaloff–Zaremba type theorem. Similar results are discussed in the settings of Riemannian manifolds and for Alexandrov surfaces. [ABSTRACT FROM AUTHOR]

In this paper, dedicated to Ireneo Peral, we study the regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy potentials in the right hand side. [ABSTRACT FROM AUTHOR]

The second and third powers of the Dirichlet kernel are used to construct discrete linear operators for the approximation of continuous periodic functions. An estimate of the rate of convergence is given. Approximation of non-periodic functions are also considered. [ABSTRACT FROM AUTHOR]

An investigation into a family of definite integrals containing log-polylog functions with negative argument will be undertaken in this paper. It will be shown that Euler sums play an important part in the solution of these integrals and some may be represented as a BBP type formula. In a special case we prove that the corresponding log integral can be represented as a linear combination of the product of zeta functions and the Dirichlet beta function. [ABSTRACT FROM AUTHOR]

Let χ be a primitive Dirichlet character modulo k ≥ 3. In this paper, we define complex numbers associated with χ, which we denote by Cr(χ)(r = 0,1, ... ), and we discuss their properties and their relationships with the generalized Bernoulli numbers. [ABSTRACT FROM AUTHOR]

The contamination game of a grid graph G(n,m) is a dynamic variant of the domination, similar to the power domination. This standard is introduced by Haynes, Hedetniemi and Henning in 2002, which is initially defined as a basic domination for a set of vertices S in a graph G, and then a propagation of this domination in all vertices of G, while starting with S. On the other hand, the contamination phenomena in G(n,m) is interpreted by an evolutionary automaton cellular, which aims to propagate viruses according to a given propagation rules. In this paper, we define a mathematical self-playing game called a contamination game based on the power domination, in which, we identify the minimum number of contaminant cells for G(n, m), called the contamination number and denoted γ (G (n,m)). [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide approximation results for space-time non-local equations with general non-local (and fractional) operators in space and time. We consider a general Markov process time changed with general subordinators or inverses to general subordinators. Our analysis is based on Bernstein symbols and Dirichlet forms, where the symbols characterize the time changes, and the Dirichlet forms characterize the Markov processes.

In this paper, we investigate hybrid power moments of generalized quadratic Gauss sums weighted with powers of Kloosterman sums and with powers of values of Dirichlet L-functions at 1. We obtain several exact formulas for prime and prime power modulus and some asymptotic formulas. [ABSTRACT FROM AUTHOR]

For any x 휖 (0, 1], let the series ∑n=1∞1/dnx be the Sylvester expansion of x, where {dj(x), j ≥ 1} is a sequence of positive integers satisfying d1(x) ≥ 2 and dj + 1(x) ≥ dj(x)(dj(x) − 1) + 1 for j ≥ 1. Suppose ϕ : ℕ → ℝ+ is a function satisfying ϕ(n+1) - ϕ (n) → ∞ as n → ∞. In this paper, we consider the setEϕ=x∈01:limn→∞logdnx−∑j=1n−1logdjxϕn=1and quantify the size of the set in the sense of Hausdorff dimension. As applications, for any β > 1 and γ > 0, we get the Hausdorff dimension of the set x∈01:limn→∞logdnx−∑j=1n−1logdjx/nβ=γ, and for any τ > 1 and η > 0, we get a lower bound of the Hausdorff dimension of the set x∈01:limn→∞logdnx−∑j=1n−1logdjx/τn=η. [ABSTRACT FROM AUTHOR]

We state a Wiener criterion at the boundary related to p-homogeneous strongly local Riemannian type Dirichlet forms.

The paper studies energy functionals on quasimetric spaces, defined by quadratic measure-valued Lagrangeans. This general model of medium, known as metric fractals, includes nested fractals and sub-Riemannian manifolds. In particular, the quadratic form of the Lagrangean satisfies Sobolev inequalities with the critical exponent determined by the (quasimetric) homogeneous dimension, which is also involved in the asymptotic distribution of the form's eigenvalues. This paper verifies that the axioms of the metric fractal are preserved by space products, leading thus to examples of non-differentiable media of arbitrary intrinsic dimension.

The aim of this paper is to study uniformly elliptic operators with general Wentzell boundary conditions in suitable $L^p$-spaces by using the approach of sesquilinear forms. We use different tools to re-prove analiticity and related results concerning the semigroups generated by the above operators. In addition, we make some complementary observations on, among other things, compactness issues and characterization of domains.

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. Modern presentations of the proof are explicitly of higher-order, in that they involve quantifying over and summing over Dirichlet characters, which are certain types of functions. The notion of a character is only implicit in Dirichlet’s original proof, and the subsequent history shows a very gradual transition to the modern mode of presentation.In this essay, we study the history of Dirichlet’s theorem with an eye towards understanding the methodological pressures that influenced some of the ontological shifts that occurred in nineteenth century mathematics. In particular, we use the history to understand some of the reasons that functions are treated as ordinary objects in contemporary mathematics, as well as some of the reasons one might want to resist such treatment. [ABSTRACT FROM AUTHOR]

Simple unsmoothed formulas to compute the Riemann zeta function, and Dirichlet L-functions to a powerfull modulus, are derived by elementary means (Taylor expansions and the geometric series). The formulas enable the square-root of the analytic conductor complexity, up to logarithmic loss, and have an explicit remainder term that is easy to control. The formula for zeta yields a convexity bound of the same strength as that from the Riemann-Siegel formula, up to a constant factor. Practical parameter choices are discussed. [ABSTRACT FROM AUTHOR]

This paper presents a novel application of Bayesian nonparametrics (BNP) for marathon data modeling. We make use of two well-known BNP priors, the single-p dependent Dirichlet process and the hierarchical Dirichlet process, in order to address two different problems. First, we study the impact of age, gender and environment on the runners’ performance. We derive a fair grading method that allows direct comparison of runners regardless of their age and gender. Unlike current grading systems, our approach is based not only on top world records, but on the performances of all runners. The presented methodology for comparison of densities can be adopted in many other applications straightforwardly, providing an interesting perspective to build dependent Dirichlet processes. Second, we analyze the running patterns of the marathoners in time, obtaining information that can be valuable for training purposes. We also show that these running patterns can be used to predict finishing time given intermediate interval measurements. We apply our models to New York City, Boston and London marathons. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of a strong solution for a parabolic singular equation in which we combine Dirichlet with integral boundary conditions given only on parts of the boundary. The proof uses a priori estimate and the density of the range of the operator generated by the problem considered. [ABSTRACT FROM AUTHOR]

The Evans-Vasilesco Theorem is proved for a general class of Dirichlet forms of diffusion type. This result is applied to show that every diffuse Radon measure vanishing on all sets of capacity zero can be expressed as the product of a Borel function and a Kato measure.

We derive a point-wise estimate for a map u : Ω ⊂ ℝn → ℝm that minimizes JA(v) : ƒA 1/2|∇v|² + U(v) subjected to the Dirichlet condition v = u on ∂Ω for every open smooth and bounded set A ⊂ Ω. We discuss some consequences of this basic estimate. [ABSTRACT FROM AUTHOR]

In this paper we establish relaxation of an arbitrary 1-equivariant wave map from $${\mathbb{R}^{1+3}_{t,x}{\setminus} (\mathbb{R}\times B(0,1))\to S^3}$$ of finite energy and with a Dirichlet condition at r = 1, to the unique stationary harmonic map in its degree class. This settles a recent conjecture of Bizoń, Chmaj, Maliborski who observed this asymptotic behavior numerically. [ABSTRACT FROM AUTHOR]

Abstract: We investigate existence and uniqueness of solutions to the filtration equation with an inhomogeneous density in ( ), approaching at infinity a given continuous datum of Dirichlet type. [Copyright &y& Elsevier]

Automatic text categorization is one of the key techniques in information retrieval and the data mining field. The classification is usually time-consuming when the training dataset is large and high-dimensional. Many methods have been proposed to solve this problem, but few can achieve satisfactory efficiency. In this paper, we present a method which combines the Latent Dirichlet Allocation (LDA) algorithm and the Support Vector Machine (SVM). LDA is first used to generate reduced dimensional representation of topics as feature in VSM. It is able to reduce features dramatically but keeps the necessary semantic information. The Support Vector Machine (SVM) is then employed to classify the data based on the generated features. We evaluate the algorithm on 20 Newsgroups and Reuters-21578 datasets, respectively. The experimental results show that the classification based on our proposed LDA+SVM model achieves high performance in terms of precision, recall and F1 measure. Further, it can achieve this within a much shorter time-frame. Our process improves greatly upon the previous work in this field and displays strong potential to achieve a streamlined classification process for a wide range of applications. [ABSTRACT FROM AUTHOR]

Let χ be a non-real Dirichlet character modulo a prime q. In this paper we prove that the distribution of the short character sum Sχ,H(x) = ∑x

Let p be a prime, χ denote the Dirichlet character modulo p and $L(p)=\{a\in {\mathbb{Z}}^{+}|(a,p)=1,a\bar{a}\equiv 1(\bmod\ p),|a-\bar{a}|\le H\}$. We study the distribution of elements in the set L( p) in character over the short interval. In this paper, we use the analytic method and show the distribution property of and give a non-trivial estimate. [ABSTRACT FROM AUTHOR]

We study the third moment of quadratic Dirichlet $$L$$-functions, obtaining an error term of size $$O(X^{3/4 + \varepsilon })$$. [ABSTRACT FROM AUTHOR]

The article presents a study regarding the trajectory attractors of autonomous and nonautonomous reaction-diffusion systems with diffusion coefficients δ ≡ 0 and δ∞(t) as t → + ¹, respectively. It highlights several elements and properties of the reaction-diffusion system which include the Laplace operator, Dirichlet conditions and the Cauchy problem. It states that the trajectory attractor of both diffusion-reaction system coincides with one another. It mentions that the study is supported by the Russian Foundation for Basic Research.

Abstract: For every positive integer n, let be the set of primitive Dirichlet characters modulo n. We show that if the Riemann hypothesis is true, then the inequality holds for all , where is the product of the first k primes, γ is the Euler–Mascheroni constant, is the twin prime constant, and is the Euler function. On the other hand, if the Riemann hypothesis is false, then there are infinitely many k for which the same inequality holds and infinitely many k for which it fails to hold. [Copyright &y& Elsevier]

Starting with finite Markov chains on partitions of a natural number n we construct, via a scaling limit transition as n → ∞, a family of infinite-dimensional diffusion processes. The limit processes are ergodic; their stationary distributions, the so-called z-measures, appeared earlier in the problem of harmonic analysis for the infinite symmetric group. The generators of the processes are explicitly described. [ABSTRACT FROM AUTHOR]

It is still a challenge to model swelling phenomena occurring in charged hydrated porous media. This is not only due to the overall complexity of the model but also to the fact that boundary conditions occur, which depend on internal variables. In the present contribution, a multi-component model based on the Theory of Porous Media (TPM) is presented. The advantage of this model is that it is thermodynamically consistent and it consists of only three primary variables. As a result of the boundary conditions depending on internal variables, the numerical treatment within the finite element method (FEM) by use of the mixed finite element scheme reveals artificial oscillations in the numerical results. To overcome these oscillations, we propose to fulfil boundary conditions weakly. [ABSTRACT FROM AUTHOR]

A method for ranking of alternatives or objects and its extensions by incomplete pairwise comparisons using random set theory are proposed in the paper. The main feature of the method is that it allows us to deal with comparisons of arbitrary groups of alternatives. The method is extended on the case of independent groups of experts. The imprecise Dirichlet model is also used to make cautious decisions in several cases. Various numerical examples illustrate the proposed method and its extensions. [ABSTRACT FROM AUTHOR]

A description of the properties of L-series with complex characters is given. By using these, together with the more familiar L-series with real characters, it is shown how certain two-dimensional lattice sums, which previously could not be put into closed form, may now be expressed in this way. [ABSTRACT FROM AUTHOR]

In this article, we investigate the behaviour of conformal capacities of several plane condensers and logarithmic capacities of some compact sets under the following types of geometric transformations: a gap moving in one or two plates, translation, and rotation of plates and breaking plates. Main motivating examples are provided by parallel plate condensers although the majority of our results are true for more general condensers. We prove monotonicity theorems in most cases and find extremal configurations with maximal capacity in more difficult ones. Our main tools are polarization and direct application of the extended Dirichlet principle. The last section of this article contains several open problems and conjectures. [ABSTRACT FROM AUTHOR]

This work establishes the local exact null controllability of non-linear parabolic integrodifferential equations with non-linearities under the integral sign in a bounded domain [image omitted] and with homogeneous Dirichlet boundary conditions. The control is assumed to be distributed along a subdomain ω of Ω. Using a Carleman type inequality we first prove the global exact null controllability of a linearized equation. Then, by a fixed point method, we obtain the main result for the non-linear case. This result asserts that, when the non-linearity is of class C1 and globally Lipschitz, the system is null controllable. [ABSTRACT FROM AUTHOR]

In this paper we consider the three-dimensional Navier-Stokes equations in infinite channel. We provide a regularity criterion for solutions of the three-dimensional Navier-Stokes equations in terms of the vertical component of the velocity field. [ABSTRACT FROM AUTHOR]

We define a notion of Kato class of measures relative to a Riemannian strongly local phomogeneous Dirichlet form and we prove a Harnack inequality (on balls that are small enough) for the positive solutions to a Schrödinger-type problem relative to the form with a potential in the Kato class. [ABSTRACT FROM AUTHOR]

We consider a reaction-diffusion-convection equation on the halfline (0,∞) with the zero Dirichlet boundary condition at x = 0. We find a positive selfsimilar solution u which blows up in a finite time T at x = 0 while u(x, T ) remains bounded for x > 0. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with a cooperative two-species Lotka–Volterra model. Using the fixed point theorem, the existence results of solutions to a strongly coupled elliptic system with homogeneous Dirichlet boundary conditions are obtained. Our results show that this model possesses at least one coexistence state if the birth rates are big and cross-diffusions are suitably weak. [Copyright &y& Elsevier]

This paper deals with the local exact boundary controllability for dynamics governed by nonlinear wave equations, subject to Dirichlet, Neumann, or any other kind of boundary controls which result in well-posedness of the corresponding initial-boundary value problem. A constructive method is developed. The local exact boundary controllability for semilinear wave equations is constructed in the case of both three (odd) and two (even) space dimensions, and the boundary control is time optimal when the space dimension is three (odd). Especially, the local exact boundary controllability is established for quasi-linear wave equations in several space dimensions by using the constructive method. [ABSTRACT FROM AUTHOR]

We consider the boundary controllability problem for a nonlinear Korteweg—de Vries equation with the Dirichlet boundary condition. We study this problem for a spatial domain with a critical length for which the linearized control system is not controllable. In order to deal with the nonlinearity, we use a power series expansion of second order. We prove that the nonlinear term gives the local exact controllability around the origin provided that the time of control is large enough. [ABSTRACT FROM AUTHOR]

We introduce and study the natural counterpart of the Dunkl Markov processes in a negatively curved setting. We give a semimartingale decomposition of the radial part, and some properties of the jumps. We prove also a law of large numbers, a central limit theorem, and the convergence of the normalized process to the Dunkl process. Eventually we describe the asymptotic behavior of the infinite loop as it was done by Anker, Bougerol and Jeulin in the symmetric spaces setting in (Iberoamericana 18: 41–97, 2002). [ABSTRACT FROM AUTHOR]

We construct an example of a maximal unimodular Dirichlet algebra whose one-point Gleason parts are dense in the maximal ideal space. The main statements were announced in [1] and now we provide their complete proofs. [ABSTRACT FROM AUTHOR]

Abstract: Using numerical methods, we will show the existence of multiple solutions for the well known logistic equation −Δu = λ g(x)u(1− u) for x ∈ Ω, with Dirichlet boundary condition. [Copyright &y& Elsevier]

Abstract: Instead of discussing the existence of a one-dimensional traveling wave front solution which connects two constant steady states, the present work deals with the case connecting a constant and a nonhomogeneous steady state on an infinite band region. The corresponding model is the well-known Fisher equation with variational coefficient and Dirichlet boundary condition. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the numerical simulation of particulate flows using a new moving mesh method combined with the multigrid fictitious boundary method (FBM) [S. Turek, D.C. Wan, L.S. Rivkind, The fictitious boundary method for the implicit treatment of Dirichlet boundary conditions with applications to incompressible flow simulations. Challenges in Scientific Computing, Lecture Notes in Computational Science and Engineering, vol. 35, Springer, Berlin, 2003, pp. 37–68; D.C. Wan, S. Turek, L.S. Rivkind, An efficient multigrid FEM solution technique for incompressible flow with moving rigid bodies. Numerical Mathematics and Advanced Applications, ENUMATH 2003, Springer, Berlin, 2004, pp. 844–853; D.C. Wan, S. Turek, Direct numerical simulation of particulate flow via multigrid FEM techniques and the fictitious boundary method, Int. J. Numer. Method Fluids 51 (2006) 531–566]. With this approach, the mesh is dynamically relocated through a (linear) partial differential equation to capture the surface of the moving particles with a relatively small number of grid points. The complete system is realized by solving the mesh movement and the partial differential equations of the flow problem alternately via an operator-splitting approach. The flow is computed by a special ALE formulation with a multigrid finite element solver, and the solid particles are allowed to move freely through the computational mesh which is adaptively aligned by the moving mesh method in every time step. One important aspect is that the data structure of the undeformed initial mesh, in many cases a tensor-product mesh or a semi-structured grid consisting of many tensor-product meshes, is preserved, while only the spacing between the grid points is adapted in each time step so that the high efficiency of structured meshes can be exploited. Numerical results demonstrate that the interaction between the fluid and the particles can be accurately and efficiently handled by the presented method. It is also shown that the presented method significantly improves the accuracy of the previous multigrid FBM to simulate particulate flows with many moving rigid particles. [Copyright &y& Elsevier]

The article presents a discussion about an analysis to a solution regarding a broad class of Dirichlet series and show the results in action in a variety of ways. It tells of an acquired integral evaluation that transpires to be a case of a rather pretty class of evaluations and worth creating explicits. It tells of a formal Dirichlet series, with certain real coefficients for other largely standard notations. It offers a solution for alternating zeta-functions, and explains that such functions can be solved by contour integration and consulting a computer algebra system.