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Cilj ovog rada je obraditi i proučiti dodatne teme iz matematike koje se pojavljuju na matematičkim natjecanjima, ali se ne obrađuju na redovnoj nastavi. Rad je zamišljen kao radni priručnik za dodatnu nastavu. Svaka tema se prvo opisuje, potom se rješava nekoliko zadataka s primjenom teme te se potom obrađuju zadaci s natjecanja. Na početku ćemo reći ponešto o samoj pripremi i edukaciji učitelja za dodatnu nastavu. Predložit ćemo i mogućnosti poboljšanja pripreme učitelja. Potom ćemo se upoznati s metodom uzastopnih približavanja te Gaussovom dosjetkom. Naučit ćemo što je to Dirichletov princip, te kako ga primjenjujemo. Pojasnit ćemo i što su diofantske jednadžbe te gdje ih koristimo, a na kraju ćemo se upoznati s logičkim i kombinatornim zadacima te zadacima s prebrojavanjima., The goal of this master thesis is to interpret and study additional topics in mathematics that appear in mathematics competitions but are not covered in regular classes. A thesis is intended to serve as a teacher’s handbook for advanced mathematics classes. Each topic is described, and the description is followed by several solved tasks with the application of the topic and eventually the tasks from the competitions are presented and solved. At the beginning, we will say something about the preparation and education of teachers for teaching in advanced classes. We will also give suggestions on how to improve teacher training. Then we will learn about the method of successive approximations and Gauss's Trick. We will learn what the Dirichlet principle is, and how we apply it. We will also explain what Diophantine equations are and where we use them, and at the end we will get acquainted with logical and combinatorial tasks and tasks with counting.

We are concerned with the Dirichlet energy of mappings defined on domains in the complex plane. The Dirichlet Principle, the name coined by Riemann, tells us that the outer variation of a harmonic mapping increases its energy. Surprisingly, when one jumps into details about inner variations, which are just a change of independent variables, new equations and related questions start to matter. The inner variational equation, called the Hopf–Laplace equation, is no longer the Laplace equation. Its solutions are generally not harmonic; we refer to them as Hopf harmonics. The natural question that arises is how does a change of variables in the domain of a Hopf harmonic map affect its energy? We show, among other results, that in case of a simply connected domain the energy increases. This should be viewed as Riemann’s Dirichlet Principle for Hopf harmonics. The Dirichlet Principle for Hopf harmonics in domains of higher connectivity is not completely solved. What complicates the matter is the insufficient knowledge of global structure of trajectories of the associated Hopf quadratic differentials, mainly because of the presence of recurrent trajectories. Nevertheless, we have established the Dirichlet Principle whenever the Hopf differential admits closed trajectories and crosscuts. Regardless of these assumptions, we established the so-called Infinitesimal Dirichlet Principle for all domains and all Hopf harmonics. Precisely, the second order term of inner variation of a Hopf harmonic map is always nonnegative.

Minimization of the Dirichlet eigenvalues of the Laplacian among sets of prescribed measure is a standard problem in shape optimization. The main result of this paper is that in the Euclidean plane, apart from the first four, no Dirichlet eigenvalue can be minimized by disks or disjoint unions of disks.

U ovom diplomskom radu obrazložene su neke metode koje se koriste pri rješavajnu kombinatornih zadataka s natjecanja u osnovnim i srednjim školama. U prvom poglavlju obrađeni su zadatci koji se rješavaju pomoću Dirichletovog principa, u drugom poglavlju zadatci koji se rješavaju pomoću Formule uključivanja-isključivanja, a u trećem poglavlju zadatci koji se rješavaju pomoću osnonovnih prinicpa prebrojavanja. This graduate thesis explains certain techniques used in solving combinatorial problems in elementary and high school competitions. In the first chapter we study the use of Dirichlet’s principle, in the second the use of the inclusion-exclusion principle, and in the third chapter we study the use of basic counting methods in problems in mathematical competitions.

In the 1870s, Carl Neumann proposed the so-called method of the arithmetic mean for solving the Dirichlet problem on convex domains. Neumann’s approach was considered at the time to be a reliable existence proof, following Weierstrass’s criticism of the Dirichlet principle. However, in 1937 H. Lebesgue pointed out a serious gap in Neumann’s proof. Curiously, the erroneous argument once again involved confusion between the notions of infimum and minimum. The objective of this paper is to show that Lebesgue’s sharp criticism of Neumann’s proof was only partially justified.

A representation formula for solutions of stochastic partial differential equations with Dirichlet boundary conditions is proved. The scope of our setting is wide enough to cover the general situation when the backward characteristics that appear in the usual formulation are not even defined in the Ito sense.

The Dirichlet principle is introduced, and early attempts on the Dirichlet problem are described; Green’s functions are introduced.

We present the theory of the Dirichlet problem for nonlocal operators which are the generators of general pure-jump symmetric Lévy processes whose Lévy measures need not be absolutely continuous. We establish basic facts about the Sobolev spaces for such operators, in particular we prove the existence and uniqueness of weak solutions. We present strong and weak variants of maximum principle, and $L^\infty$ bounds for solutions. We also discuss the related extension problem in $C^{1,1}$ domains.

We discuss some local analytic properties of the ring of Dirichlet series. We obtain mainly the equivalence between the irreducibility in the analytic ring and in the formal one. In the same way we prove that the ring of analytic Dirichlet series is integrally closed in the ring of formal Dirichlet series. Finally we introduce the notion of standard basis in these rings and we give a finitely generated ideal which does not admit standard bases.

The k-Hessian operator σ k is the k-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the k-Hessian equation σ k ( D 2 u ) = f with Dirichlet boundary condition u = 0 is variational; indeed, this problem can be studied by means of the k-Hessian energy − ∫ u σ k ( D 2 u ) . We construct a natural boundary functional which, when added to the k-Hessian energy, yields as its critical points solutions of k-Hessian equations with general non-vanishing boundary data. As a consequence, we establish a Dirichlet's principle for k-admissible functions with prescribed Dirichlet boundary data.

We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a $1$-Poincar\'e inequality. We propose a notion of \emph{domain with boundary of positive mean curvature} and prove that, for such domains, there is always a solution to the Dirichlet problem for least gradients with continuous boundary data. Here \emph{least gradient} is defined as minimizing total variation (in the sense of BV functions) and boundary conditions are satisfied in the sense that the \emph{boundary trace} of the solution exists and agrees with the given boundary data. This extends the result of Sternberg, Williams and Ziemer to the non-smooth setting. Via counterexamples we also show that uniqueness of solutions and existence of \emph{continuous} solutions can fail, even in the weighted Euclidean setting with Lipschitz weights.

We study conformal deformation problems on manifolds with boundary which include prescribing σ k ≡ 0 in the interior. In particular, we prove a Dirichlet principle when the induced metric on the boundary is fixed and an Obata-type theorem on the upper hemisphere. We introduce some conformally covariant multilinear operators as a key technical tool.

We show that every linear functional on the Dirichlet space that is non-zero on nowhere-vanishing functions is necessarily a multiple of a point evaluation. Continuity of the functional is not assumed. As an application, we obtain a characterization of weighted composition operators on the Dirichlet space as being exactly those linear maps that send nowhere-vanishing functions to nowhere-vanishing functions. We also investigate possible extensions to weighted Dirichlet spaces with superharmonic weights. As part of our investigation, we are led to determine which of these spaces contain functions that map the unit disk onto the whole complex plane.

In this paper, we study the distribution of zeros of the ordinary Dirichlet polynomials which are generated by an equivalence relation introduced by Harald Bohr. Through the use of completely multiplicative functions, we construct equivalent Dirichlet polynomials which have the same critical strip, where all their zeros are situated, and satisfy the same topological property consisting of possessing zeros arbitrarily near every vertical line contained in some substrips inside their critical strip. We also show that the real projections of the zeros of the partial sums of the alternating zeta function, for some particular cases, are dense in their critical intervals. The second author's research was partially supported by Generalitat Valenciana under project GV/2015/035.

Let be a continuous function which increases to on an infinite interval of the form . A necessary and sufficient condition is found on a sequence increasing to which ensures that for each Dirichlet series of the form , , which is absolutely convergent in the following relation holds: where and are the maximum modulus and maximum term of the series, respectively. Bibliography: 10 titles.

The paper shows the unique solvability of the classical Dirichlet problem in cylindrical domain for threedimensional elliptic equations with degeneration of type and order.

We study Dirichlet problem for a nonlinear equation with a drift term. Despite the presence of the singular convection term, we establish existence and uniqueness of a solution in spaces larger than the natural one.

Z. Nehari developed a general technique for obtaining inequalities for conformal maps and domain functions from contour integrals and the Dirichlet principle. Given a harmonic function with singularity on a domain $R$, it associates a monotonic functional of subdomains $D \subseteq R$. In the case that $R$ is conformally equivalent to a disk, we extend Nehari's method by associating a functional to any quadratic differential on $R$ with specified singularities. Nehari's method corresponds to the special case that the quadratic differential is of the form $(\partial q)^2$ for a singular harmonic function $q$ on $R$. Besides being more general, our formulation is conformally invariant, and has a particularly elegant equality statement. As an application we give a one-parameter family of monotonic, conformally invariant functionals which correspond to growth theorems for bounded univalent functions. These generalize and interpolate the Pick growth theorems, which appear in a conformally invariant form equivalent to a two-point distortion theorem of W. Ma and D. Minda., Comment: 31 pages

In this paper, a new formulation is proposed to evaluate the origin intensity factors (OIFs) in the singular boundary method (SBM) for solving 3D potential problems with Dirichlet boundary condition. The SBM is a strong-form boundary discretization collocation technique and is mathematically simple, easy-to-program, and free of mesh. The crucial step in the implementation of the SBM is to determine the OIFs which isolate the singularities of the fundamental solutions. Traditionally, the inverse interpolation technique (IIT) is adopted to calculate the OIFs on Dirichlet boundary, which is time consuming for large-scale simulation. In recent years, the new methodology has been developed to efficiently calculate the OIFs on Neumann boundary, but the Dirichlet problem remains an open issue. This study employs the subtracting and adding-back technique based on the integration of the fundamental solution over the whole boundary to develop a new formulation of the OIFs on 3D Dirichlet boundary. Several problems with varied domain shapes and boundary conditions are carried out to validate the effectiveness and feasibility of the proposed scheme in comparison with the SBM based on inverse interpolation technique, the method of fundamental solutions, and the boundary element method.

We study the Dirichlet problem of the n-dimensional complex Monge—Ampere equation det(u ij ) = F/|z|2α, where 0 < α < n. This equation comes from La Nave—Tian’s continuity approach to the Analytic Minimal Model Program.

We investigate the Dirichlet problem for the telegraph equation in a rectangular domain. We establish a criterion of uniqueness of solution to the problem. The solution is constructed as the sum of an orthogonal series. In substantiation of convergence of the series, the problem of small denominators occurs. In connection with this, we establish estimates ensuring separation from zero of denominators with the corresponding asymptotics which allow us to prove the existence of a regular solution and prove its stability under small perturbations of boundary functions.

In this paper, we study the existence of at least three distinct solutions for a perturbed anisotropic discrete Dirichlet problem. Our approach is based on recent variational methods for smooth functionals defined on reflexive Banach spaces. Some examples are presented to demonstrate the application of our main results.

There exist many explicit evaluations of Dirichlet series. Most of them are constructed via the same approach: by taking products or powers of Dirichlet series with a known Euler product representation. In this paper we derive a result of a new flavour: we give the Dirichlet series representation to solution $f=f(s,w)$ of the functional equation $L(s-wf)=\exp(f)$, where $L(s)$ is the L-function corresponding to a completely multiplicative function. Our result seems to be a Dirichlet series analogue of the well known Lagrange-B\"urmann formula for power series. The proof is probabilistic in nature and is based on Kendall's identity, which arises in the fluctuation theory of L\'evy processes., Comment: 12 pages, 1 figure

We prove compactness and existence results for a scalar curvature-type equation on a bounded domain of . The problem has a variational structure with lack of compactness. Using an asymptotic analysis and dynamical arguments, we characterize the accumulation points of all non-compact flow-lines of the gradient vector field, the so-called critical points at infinity. We then associate to each critical point at infinity a Morse index, which enables us to derive a Bahri–Coron index type criteria and to establish the existence.

In this paper, we investigate the existence and the asymptotic behaviour of positive classical solutions for the following non-linear problem where and a is a function in , such that there exist satisfying for each , where and for , , , with is a continuous function on such that . The function g is a nonnegative in such that there exist satisfying for and for , where .

With the help of transformation formulas of Dirichlet L-series, we generalize some classical formulas for the values $$\zeta (2N+1)$$ given by Ramanujan. This will be done by constructing generalized Dirichlet series of the form $$\sum \nolimits _{n=1}^\infty a_n n^{-s/b}$$ where $$b > 0$$ is an integer, which have similar transformation properties as Dirichlet L-functions and by considering their Mellin transforms using contour integration methods.

A k variable integral is introduced, which will produce the total integral eqivalent to the total integral in a $$(k-1)$$ -variate type-1 Dirichlet integral. Hence this author calls the new integral as type-1 pseudo Dirchilet integral. A statistical density is introduced based on this pseudo Dirichlet integral, which will be called the type-1 pseudo Dirichlet density. Marginal densities, moments and other properties are studied. Then several transformations are considered which will give different forms of the type-1 pseudo Dirichlet density. Then the corresponding real matrix-variate situation is studied and the real matrix-variate type-1 pseudo Dirichlet density is introduced. Several transformations in the matrix-variate case are also discussed which will give different representations of the proposed density and finally leading to a real type-1 Dirichlet density.

Triple solutions are obtained for a discrete problem involving a nonlinearly perturbed one-dimensional p(k)-Laplacian operator and satisfying Dirichlet boundary conditions. The methods for existence rely on a Ricceri-local minimum theorem for differentiable functionals. Several examples are included to illustrate the main results.

On the Hilbert space of Dirichlet series with square summable coefficients, there are no non-normal complex symmetric composition operators induced by non-constant symbols. This is in sharp contrast with the phenomenon on the classical Hardy space over the unit disk.

Let O ⊂ R d be a bounded domain of class C 1,1. Let 0 < e - 1. In L 2(O;C n ) we consider a positive definite strongly elliptic second-order operator B D,e with Dirichlet boundary condition. Its coefficients are periodic and depend on x/e. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent (B D,e − ζQ 0(·/e))−1 as e → 0. Here the matrix-valued function Q 0 is periodic, bounded, and positive definite; ζ is a complex-valued parameter. We find approximations of the generalized resolvent in the L 2(O;C n )-operator norm and in the norm of operators acting from L 2(O;C n ) to the Sobolev space H 1(O;C n ) with two-parameter error estimates (depending on e and ζ). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation Q 0(x/e)∂ t v e (x, t) = −(B D,e v e )(x, t).

In this paper we consider a class of Dirichlet series with multiplicative coefficients which define functions holomorphic in the right half of the complex plane, and for which there are sequences of Dirichlet polynomials that converge uniformly to these functions in any rectangle within the critical strip. We call such polynomials approximating Dirichlet polynomials. We study the properties of the approximating polynomials, in particular, for those Dirichlet series, whose coefficients are determined by nonprincipal generalized characters, i.e. finite-valued numerical characters which do not vanish on almost all prime numbers and whose summatory function is bounded. These developments are interesting in connection with the problem of the analytical continuation of such Dirichlet series to the entire complex plane, which in turn is tied with the solution of a well-known Chudakov hypothesis about every generalized character being a Dirichlet character.

In this paper, by using the Beurling-Nevanlinna type inequality we obtain new results on the existence of solutions of the Dirichlet problem with respect to the Schrödinger-prey operator. Meanwhile, the local stability of the Schrödingerean equilibrium and endemic equilibrium of the model are also discussed. We specially analyze the existence and stability of the Schrödingerean Hopf bifurcation by using the center manifold theorem and bifurcation theory. As applications, theoretic analysis and numerical simulation show that the Schrödinger-prey system with latent period has a very rich dynamic characteristics.

An energy space based Dirichlet boundary control problem governed by biharmonic equation is investigated and subsequently a C0-interior penalty method is proposed and analyzed. An abstract a priori error estimate is derived under the minimal regularity conditions. The abstract error estimate guarantees optimal order of convergence whenever the solution is sufficiently regular. Further an optimal order L2-norm error estimate is derived. Numerical experiments illustrate the theoretical findings.

The present paper mainly gives some applications of Berezin type symbols on the Dirichlet space of unit ball. We study the solvability of some Riccati operator equations of the form XAX + XB - CX = D related to harmonic Toeplitz operators on the Dirichlet space. Especially, the invariant subspaces of Toeplitz operators are also considered.

We obtain sufficient conditions for the solvability of the Dirichlet problem for a mixed-type equation of the second kind in a rectangular domain. The solution is represented by a convergent series constructed from the problem data. Some cases of nonuniqueness of the solution are described.

We study the behavior near the boundary conical point of weak solutions to the Dirichlet problem for elliptic quasi-linear second-order equation with the p -Laplacian and the strong nonlinearity on the right side.

We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold M of dimension n 2 for a large class of operators containing, in particular, the p-Laplacian and the minimal graph operator. We extend several existence results obtained for the p-Laplacian to our class of operators. As an application of our main result, we prove the solvability of the asymptotic Dirichlet problem for the minimal graph equation for any continuous boundary data on a (possibly non rotationally symmetric) manifold whose sectional curvatures are allowed to decay to 0 quadratically.