Cusp forms in S 6(Γ 0(23)), S 8(Γ 0(23)) and the number of representations of numbers by some quadratic forms in 12 and 16 variables

In this study we find bases of S ₆(Γ ₀(23)), S ₈(Γ ₀(23)) and obtain explicit formulae for the number of representations of numbers by some quadratic forms in 12 and 16 variables that are direct sums of binary quadratic forms $F_{1}=x_{1}^{2}+x_{1}x_{2}+6x_{2}^{2}$ and $\varPhi_{1}=2x_{1}^{2}+x_{1}x_{2}+3x_{2}^{2}$ (or its inverse) with discriminant ?23.     

Macdonald operators at infinity

We construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables x ₁,x ₂,… and of two parameters q,t are their eigenfunctions. These operators are defined as limits at N→∞ of renormalized Macdonald operators acting on symmetric polynomials in the variables x ₁,…,x _N . They are differential operators in terms of the power sum variables \(p_{n}=x_{1}^{n}+x_{2}^{n}+\cdots\) and we compute their symbols by using the Macdonald reproducing kernel. We express these symbols in terms of the Hall–Littlewood symmetric functions of the variables x ₁,x ₂,…. Our result also yields elementary step operators for the Macdonald symmetric functions.     

High Frequency Resolvent Estimates for Perturbations by Large Long-range Magnetic Potentials and Applications to Dispersive Estimates

We prove optimal high-frequency resolvent estimates for self-adjoint operators of the form ${G=\left(i\nabla+b(x)\right)^2+V(x)}$ on ${L^2({\bf R}^n), n\ge 3}$ , where the magnetic potential b(x) and the electric potential V(x) are long-range and large. As an application, we prove dispersive estimates for the wave group ${{\rm e}^{it\sqrt{G}}}$ in the case n = 3 for potentials b(x), V(x) = O(|x|^?2-δ ) for ${|x|\gg 1}$ , where δ > 0.     

The asymptotic behavior of the spectral function for elliptic operators in an unbounded region

We consider elliptic self-adjoint differential operators L of order 2m in a bounded region D? R_n. An asymptotic formula for the function N(λ) = \(N(\lambda ) = \sum\limits_{\lambda _n< \lambda } 1 \) the number of eigenvalues of the operator L less than A. is proved: $$N(\lambda ) = M_0 \lambda ^{n/2m} + o(\lambda ^{n/2m} )$$ whereλ → + ∞ and M₀ is the following constant: $$M_0 = \frac{{V_D }}{{(2\pi )^n \Gamma (1 + n/2m)}}\int_{R_n } {e^{ - L(s)} ds} .$$      

Kolmogorov-type inequalities and the best formulas for numerical differentiation

For a certain class of complex-valued functionsf(x), ?∞ $$u_N = \mathop {\inf }\limits_{\parallel A\parallel \leqslant N_\parallel f^{(n)} \parallel _{L_2 \leqslant } 1} \parallel f^{(k)} - A(f)\parallel C$$ of a differential operator by linear operators A with the norm ∥A∥ _L2 ^C ≤N,N,>0. Using the value u_N, the smallest constant Q in the inequality $$\parallel f^{(k)} \parallel _Q \leqslant Q\parallel f\parallel _{L_2 }^\alpha \parallel f^{(n)} \parallel _{L_2 }^\beta $$ is found.     

Joint Spectral Multipliers for Mixed Systems of Operators

We obtain a general Marcinkiewicz-type multiplier theorem for mixed systems of strongly commuting operators \(L=(L_1,\ldots ,L_d);\) where some of the operators in L have only a holomorphic functional calculus, while others have additionally a Marcinkiewicz-type functional calculus. Moreover, we prove that specific Laplace transform type multipliers of the pair \((\mathcal {L},A)\) are of certain weak type (1, 1). Here \(\mathcal {L}\) is the Ornstein-Uhlenbeck operator while A is a non-negative operator having Gaussian bounds for its heat kernel. Our results include the Riesz transforms \(A(\mathcal {L}+A)^{-1},\) \(\mathcal {L}(\mathcal {L}+A)^{-1}\).     

Nikodym Maximal Functions Associated with Variable Planes in {\mathbb{R}^{3}}

Given a vector field ${\mathfrak{a}}$ on ${\mathbb{R}^3}$ , we consider a mapping ${x\mapsto \Pi_{\mathfrak{a}}(x)}$ that assigns to each ${x\in\mathbb{R}^3}$ , a plane ${\Pi_{\mathfrak{a}}(x)}$ containing x, whose normal vector is ${\mathfrak{a}(x)}$ . Associated with this mapping, we define a maximal operator ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^1_{loc}(\mathbb{R}^3)}$ for each ${N\gg 1}$ by $$\mathcal{M}^{\mathfrak{a}}_Nf(x)=\sup_{x\in\tau} \frac{1}{|\tau|} \int_{\tau}|f(y)|\,dy$$ where the supremum is taken over all 1/N ×? 1/N?× 1 tubes τ whose axis is embedded in the plane ${\Pi_\mathfrak{a}(x)}$ . We study the behavior of ${\mathcal{M}^{\mathfrak{a}}_N}$ according to various vector fields ${\mathfrak{a}}$ . In particular, we classify the operator norms of ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^2(\mathbb{R}^3)}$ when ${\mathfrak{a}(x)}$ is the linear function of the form (a ₁₁ x ₁?+?a ₂₁ x ₂, a ₁₂ x ₁?+?a ₂₂ x ₂, 1). The operator norm of ${\mathcal{M}^\mathfrak{a}_N}$ on ${L^2(\mathbb{R}^3)}$ is related with the number given by $$D=(a_{12}+a_{21})^2-4a_{11}a_{22}.$$      

An improvement on Waring?CGoldbach problem for unlike powers

Let p _i be prime numbers. In this paper, it is proved that for any integer k?R5, with at most $O\big(N^{1-\frac{1}{3k\times2^{k-2}}+\varepsilon}\big)$ exceptions, all positive even integers up to N can be expressed in the form $p_{2}^{2}+p_{3}^{3}+p_{5}^{5}+p_{k}^{k}$ . This improves the result $O\big(\frac{N}{\log^{c}N}\big)$ for some c>0 due to Lu and Shan [12], and it is a generalization for a series of results of Ren and Tsang [15], [16] and Bauer [1?C4] for the problem in the form $p_{2}^{2}+p_{3}^{3}+p_{4}^{4}+p_{5}^{5}$ . This method can also be used for some other similar forms.     

Algebraic and geometric properties of quadratic Hamiltonians determined by sectional operators

Following the terminology introduced by V. V. Trofimov and A. T. Fomenko, we say that a self-adjoint operator $\varphi :\mathfrak{g}* \to \mathfrak{g}$ is sectional if it satisfies the identity ad _?x ^* a = ad _β ^* x, $x \in \mathfrak{g}*$ , where $\mathfrak{g}$ is a finite-dimensional Lie algebra and $a \in \mathfrak{g}*$ and $\beta \in \mathfrak{g}$ are fixed elements. In the case of a semisimple Lie algebra $\mathfrak{g}$ , the above identity takes the form [?x, a] = [β, x] and naturally arises in the theory of integrable systems and differential geometry (namely, in the dynamics of n-dimensional rigid bodies, the argument shift method, and the classification of projectively equivalent Riemannian metrics). This paper studies general properties of sectional operators, in particular, integrability and the bi-Hamiltonian property for the corresponding Euler equation $\dot x = ad_{\varphi x}^* x$ .     

Long-time asymptotics for fully nonlinear homogeneous parabolic equations

We study the long-time asymptotics of solutions of the uniformly parabolic equation $$ u_t + F(D^2u) = 0 \quad{\rm in}\, {\mathbb{R}^{n}}\times \mathbb{R}_{+},$$ for a positively homogeneous operator F, subject to the initial condition u(x, 0) =  g(x), under the assumption that g does not change sign and possesses sufficient decay at infinity. We prove the existence of a unique positive solution Φ⁺ and negative solution Φ^?, which satisfy the self-similarity relations $$\Phi^\pm (x,t) = \lambda^{\alpha^\pm}\Phi^\pm ( \lambda^{1/2} x,\lambda t ).$$ We prove that the rescaled limit of the solution of the Cauchy problem with nonnegative (nonpositive) initial data converges to ${\Phi^+}$ ( ${\Phi^-}$ ) locally uniformly in ${\mathbb{R}^{n} \times \mathbb{R}_{+}}$ . The anomalous exponents α⁺ and α^? are identified as the principal half-eigenvalues of a certain elliptic operator associated to F in ${\mathbb{R}^{n}}$ .     

Hermitian generalization of the Rarita-Schwinger operators

We introduce two new linear differential operators which are invariant with respect to the unitary group SU（n）. They constitute analogues of the twistor and the Rarita-Schwinger operator in the orthogonal case. The natural setting for doing this is Hermitian Clifford Analysis. Such operators are constructed by twisting the two versions of the Hermitian Dirac operator 6z_ and 6z_ and then projecting on irreducible modules for the unitary group. We then study some properties of their spaces of nullsolutions and we find a formulation of the Hermitian Rarita-Schwinger operators in terms of Hermitian monogenic polynomials.     

Relative widths of function classes of L 2(T) defined by a linear differential operator in L q (T)

In this paper, the smallest number M which makes the equality $$ K_n (W_2^{L_r } (T),MW_2^{L_r } (T),L_2 (T)) = d_n (W_2^{L_r } (T),L_2 (T)) $$ valid, is established and the asymptotic order of $$ K_n (W_2^{L_r } (T),W_2^{L_r } (T),L_q (T)),1 \leqslant q \leqslant \infty $$ , is obtained, where $ W_2^{L_r } $ (T) is a periodic smooth function class which is determined by a linear differential operator, K _n (·, ·, ·) and d _n (·, ·) are the relative width and the width in the sense of Kolmogorov, respectively.     

Representations of Positive Integers by a Direct Sum of Quadratic forms

The number of representation of positive integers by quadratic forms $ F_{1}=x_{1}^{2}+3x_{1}x_{2}+8x_{2}^{2} $ and $ G_{1}=2x_{1}^{2}+3x_{1}x_{2}+4x_{2}^{2} $ of discriminant —23 are given. Moreover, a basis for the cusp form space S ₄(Γ₀(23), 1) are constructed. Furthermore, formulas for the representation of positive integers by direct sum of copies of F ₁ and G ₁, i.e. formulas for $ r(n; F_{4}), r(n; G_{4}), r(n; F_{3} \oplus G_{1}), r(n; F_{2} \oplus G_{2}), {\rm and}\ r(n; F_{1} \oplus G_{3}) $ , are derived using the elements of the space S ₄(Γ(23), 1).     

Path integrals and the essential self-adjointness of differential operators on noncompact manifolds

We consider Schrödinger operators on possibly noncompact Riemannian manifolds, acting on sections in vector bundles, with locally square integrable potentials whose negative part is in the underlying Kato class. Using path integral methods, we prove that under geodesic completeness these differential operators are essentially self-adjoint on $\mathsf{C }^{\infty }_0$ , and that the corresponding operator closures are semibounded from below. These results apply to nonrelativistic Pauli–Dirac operators that describe the energy of Hydrogen type atoms on Riemannian $3$ -manifolds.     

On the existence and smoothness of a generalized solution of a nonlinear differential equation with degeneration

Let Ω be an arbitrary open set in R _n , and let σ(x) and g _i (x), i = 1, 2, ..., n, be positive functions in Ω. We prove a embedding theorem of different metrics for the spaces W _p ^r (Ω, σ, $ \vec g $ ), where r ∈ N, p ≥ 1, and $ \vec g $ (x) = (g ₁(x), g ₂(x), ..., g _n (x)), with the norm $$ \left\| {u;W_p^r (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\left\| {u;L_{p,r}^r (\Omega ;\sigma ,\vec g)} \right\|^p + \left\| {u;L_{p,r}^0 (\Omega ;\sigma ,\vec g)} \right\|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where $$ \left\| {u;L_{p,r}^m (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\sum\limits_{\left| k \right| = m} {\int\limits_\Omega {(\sigma (x)g_1^{k_1 - r} (x)g_2^{k_2 - r} (x) \cdots g_n^{k_n - r} (x)\left| {u^{(k)} (x)} \right|)^p dx} } } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ We use this theorem to prove the existence and uniqueness of a minimizing element U(x) ∈ W _p ^r (Ω, σ, $ \vec g $ ) for the functional $$ \Phi (u) = \sum\limits_{\left| k \right| \leqslant r} {\frac{1} {{p_k }}\int\limits_\Omega {a_k (x)} \left| {u^{(k)} (x)} \right|^{p_k } } dx - \left\langle {F,u} \right\rangle , $$ where F is a given functional. We show that the function U(x) is a generalized solution of the corresponding nonlinear differential equation. For the case in which Ω is bounded, we study the differential properties of the generalized solution depending on the smoothness of the coefficients and the right-hand side of the equation.     

A Note on Quasi-paranormal Operators

Let T be a bounded linear operator on an infinite dimensional complex Hilbert space. In this paper, we introduce the new class, denoted ${{\mathcal{QP}}}$ , of operators satisfying ${{\|T^{2}x\|^{2}\leq \|T^{3}x\|\|Tx\|}}$ for all ${{x \in \mathcal{H}}}$ . This class includes the classes of paranormal operators and quasi-class A operators. We prove basic structural properties of these operators. Using these results, we also prove that if E is the Riesz idempotent for a nonzero isolated point λ₀ of the spectrum of ${{T \in \mathcal{QP}}}$ , then E is self-adjoint if and only if ${{N(T-\lambda_{0}) \subseteq N(T^{*}-\overline{\lambda}_{0})}}$ .     

The long time error analysis in the second order difference type method of an evolutionary integral equation with completely monotonic kernel

We study convergence of the numerical methods in which the second order difference type method is combined with order two convolution quadrature for approximating the integral term of the evolutionary integral equation $$ u^{\prime}(t)+\int_{0}^{t}\beta(t-s)\,A\,u\,(s)\;ds \, =\,0 ,~~~~ t>0,~~u(0)=u_{0}, $$ which arises in the theory of linear viscoelasticity. Here A is a positive self-adjoint densely defined linear operator in a real Hilbert space H and \(\beta (t)\) is completely monotonic and locally integrable, but not constant. We establish the convergence properties of the discretization in time in the \(l_{t}^{1}(0,\infty ;\,H)\) or \(l_{t}^{\infty }(0,\infty ;\,H)\) norm.     

Classical and Sobolev orthogonality of the nonclassical Jacobi polynomials with parameters \alpha =\beta =-1

In this paper, we consider the second-order differential expression $$\begin{aligned} \ell [y](x)=(1-x^{2})(-(y^{\prime }(x))^{\prime }+k(1-x^{2})^{-1} y(x))\quad (x\in (-1,1)). \end{aligned}$$ This is the Jacobi differential expression with nonclassical parameters $\alpha =\beta =-1$ in contrast to the classical case when $\alpha ,\beta >-1$ . For fixed $k\ge 0$ and appropriate values of the spectral parameter $\lambda ,$ the equation $\ell [y]=\lambda y$ has, as in the classical case, a sequence of (Jacobi) polynomial solutions $\{P_{n}^{(-1,-1)} \}_{n=0}^{\infty }.$ These Jacobi polynomial solutions of degree $\ge 2$ form a complete orthogonal set in the Hilbert space $L^{2}((-1,1);(1-x^{2})^{-1})$ . Unlike the classical situation, every polynomial of degree one is a solution of this eigenvalue equation. Kwon and Littlejohn showed that, by careful selection of this first-degree solution, the set of polynomial solutions of degree $\ge 0$ are orthogonal with respect to a Sobolev inner product. Our main result in this paper is to construct a self-adjoint operator $T$ , generated by $\ell [\cdot ],$ in this Sobolev space that has these Jacobi polynomials as a complete orthogonal set of eigenfunctions. The classical Glazman–Krein–Naimark theory is essential in helping to construct $T$ in this Sobolev space as is the left-definite theory developed by Littlejohn and Wellman.     

On a certain class of commuting systems of linear operators

In this paper we describe the class of commuting pairs of bounded linear operators {A ₁,A ₂} acting on a Hilbert space H which are unitarily equivalent to the system of integrations over independent variables $$ (\tilde A_1 f)(x,y) = i\int_x^a {f(t,y)} dt, (\tilde A_2 f)(x,y) = i\int_y^b {f(x,s)} ds $$ in $ L_{\Omega _L }^2 $ , where ?? _L is the compact set in ? ₊ ² bounded by the lines x = a and y = b and by a decreasing smooth curve L = {((x, p(x)): p(x) ?? C _[0,a] ¹ , p(0) = b, p(a) = 0}.     

Critical points for functionals with quasilinear singular Euler–Lagrange equations

For a bounded, open subset Ω of ${\mathbb{R}^{N}}$ with N > 2, and a measurable function a(x) satisfying 0 < α ≤ a(x) ≤ β, a.e. ${x \in \Omega}$ , we study the existence of positive solutions of the Euler–Lagrange equation associated to the non-differentiable functional $$\begin{array}{ll}J(v) = \frac{1}{2} \int \limits_{\Omega} [a(x)+|v|^{\gamma}]| \nabla v|^{2}- \frac{1}{p} \int \limits_{\Omega}(v_{+})^p,\end{array}$$ if γ > 0 and p > 1. Special emphasis is placed on the case ${2^{*} < p < \frac{2^{*}}{2} ( \gamma +2 )}$ .