Mathematics - Classical Analysis and ODEs Publications (50)


Mathematics - Classical Analysis and ODEs Publications

Motivated by applications to renewal theory, Erd\"os, de Bruijn and Kingman posed in $50$th-$70$th a problem on regularity of reciprocals of probability generating functions. We solve the problem in the strong negative and give a number of other related results. Read More

We construct an a.e. approximately differentiable homeomorphism of a unit $n$-dimensional cube onto itself which is orientation preserving, has the Lusin property (N) and has the Jacobian determinant negative a. Read More

We present $\text{Fuchsia}$ $-$ an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients $\partial_x\,\mathbf{f}(x,\epsilon) = \mathbb{A}(x,\epsilon)\,\mathbf{f}(x,\epsilon)$ finds a basis transformation $\mathbb{T}(x,\epsilon)$, i.e., $\mathbf{f}(x,\epsilon) = \mathbb{T}(x,\epsilon)\,\mathbf{g}(x,\epsilon)$, such that the system turns into the epsilon form: $\partial_x\, \mathbf{g}(x,\epsilon) = \epsilon\,\mathbb{S}(x)\,\mathbf{g}(x,\epsilon)$, where $\mathbb{S}(x)$ is a Fuchsian matrix. Read More

A vector field $B$ is said to be Beltrami vector field (force free-magnetic vector field in physics), if $B\times(\nabla\times B)=0$. Motivated by our investigations on projective superflows, and as an important side result, we construct two unique Beltrami vector fields $\mathfrak{I}$ and $\mathfrak{Y}$, such that $\nabla\times\mathfrak{I}=\mathfrak{I}$, $\nabla\times\mathfrak{Y}=\mathfrak{Y}$, and that both have orientation-preserving icosahedral symmetry (group of order $60$). Analogous constructions are done for tetrahedral and octahedral groups of orders $12$ and $24$, respectively. Read More

Consider an abstract operator $L$ which acts on monomials $x^n$ according to $L x^n= \lambda_n x^n + \nu_n x^{n-2}$ for $\lambda_n$ and $\nu_n$ some coefficients. Let $P_n(x)$ be eigenpolynomials of degree $n$ of $L$: $L P_n(x) = \lambda_n P_n(x)$. A classification of all the cases for which the polynomials $P_n(x)$ are orthogonal is provided. Read More

We establish criteria for a logarithmic potential on the sub-Riemannian Heisenberg group to be comparable to the Jacobian of a quasiconformal mapping of the same. The results rest on an extension to the theory of quasiconformal flows on the Heisenberg group, and constructions that adapt the iterative method of Bonk, Heinonen, and Saksman to this setting. Finally, we initiate an exploration of the geometric consequences, using the logarithmic potentials to distort the geometry of the Heisenberg group, then proving these deformations are bi-Lipschitz equivalent to the original. Read More

In this paper, we first introduce certain forms of extended incomplete Pochhammer symbols which are then used to define families of extended incomplete generalized hypergeometric functions. For these functions, we investigate various properties including the integral representations, derivative formula, certain generating function and fractional integrals (and derivatives) relationships. Some special cases of the main results are also deduced. Read More

We give several sharp estimates for a class of combinations of second order Riesz transforms on Lie groups ${G}={G}_{x} \times {G}_{y}$ that are multiply connected, composed of a discrete abelian component ${G}_{x}$ and a connected component ${G}_{y}$ endowed with a biinvariant measure. These estimates include new sharp $L^p$ estimates via Choi type constants, depending upon the multipliers of the operator. They also include weak-type, logarithmic and exponential estimates. Read More

We consider a reconstruction problem for "spike-train" signals $F$ of the a priori known form $ F(x)=\sum_{j=1}^{d}a_{j}\delta\left(x-x_{j}\right),$ from their moments $m_k(F)=\int x^kF(x)dx.$ We assume $m_k(F)$ to be known for $k=0,1,\ldots,2d-1,$ with an absolute error not exceeding $\epsilon > 0$. This problem is essentially equivalent to solving the Prony system $\sum_{j=1}^d a_jx_j^k=m_k(F), \ k=0,1,\ldots,2d-1. Read More

We prove finite-time singularity formation for De Gregorio's model of the three-dimensional vorticity equation in the class of $L^p\cap C^\alpha(\mathbb{R})$ vorticities for some $\alpha>0$ and $p<\infty$. We also prove finite-time singularity formation from smooth initial data for the Okamoto-Sakajo-Wunsch models in a new range of parameter values. As a consequence, we have finite-time singularity for certain infinite-energy solutions of the surface quasi-geostrophic equation which are $C^\alpha$-regular. Read More

We derive solvability conditions and closed-form solution for the Weber type integral equation, related to the familiar Weber-Orr integral transforms and the old Weber-Titchmarsh problem (posed in Proc. Lond. Math. Read More

Our aim in this paper is to show some new inequalities for Mathieu's type series and Riemann zeta function. In particular, some Tur\'an type inequalities, some monotonicity and log-convexity results for these special functions are given. New Laplace type integral representations for Mathie type series and Riemann zeta function are also presented. Read More

We express a certain complex-valued solution of Legendre's differential equation as the product of an oscillatory exponential function and an integral involving only nonoscillatory elementary functions. By calculating the logarithmic derivative of this solution, we show that Legendre's differential equation admits a nonoscillatory phase function. Moreover, we derive from our expression an asymptotic expansion useful for evaluating Legendre functions of the first and second kinds of large orders, as well as the derivative of the nonoscillatory phase function. Read More

After a short review of the classical Lie theorem, a finite dimensional Lie algebra of vector fields is considered and the most general conditions under which the integral curves of one of the fields can be obtained by quadratures in a prescribed way will be discussed, determining also the number of quadratures needed to integrate the system. The theory will be illustrated with examples andbn an extension of the theorem where the Lie algebras are replaced by some distributions will also be presented. Read More

The classical Lebedev index transform (1967), involving squares and products of the Legendre functions is generalized on the associated Legendre functions of an arbitrary order. Mapping properties are investigated in the Lebesgue spaces. Inversion formulas are proved. Read More

In this paper, we investigate optimal linear approximations ($n$-approximation numbers ) of the embeddings from the Sobolev spaces $H^r\ (r>0)$ for various equivalent norms and the Gevrey type spaces $G^{\alpha,\beta}\ (\alpha,\beta>0)$ on the sphere $\Bbb S^d$ and on the ball $\Bbb B^d$, where the approximation error is measured in the $L_2$-norm. We obtain preasymptotics, asymptotics, and strong equivalences of the above approximation numbers as a function in $n$ and the dimension $d$. We emphasis that all equivalence constants in the above preasymptotics and asymptotics are independent of the dimension $d$ and $n$. Read More

The main purpose of this note is to increase awareness of the recent emergence of the polynomial method as a powerful tool in restriction theory. I hope this note will initiate fruitful discussions during the forthcoming MSRI program on harmonic analysis and will lead to progress on important problems. The main observation presented here is that breaking the $\frac32$ barrier in Wolff's Kakeya estimate in $\R^4$ leads to improving the $\frac{14}{5}$ threshold for the related restriction problem for the paraboloid. Read More

We will discuss a somewhat striking spectral property of finitely valued stationary processes on Z that says that if the spectral measure of the process has a gap then the process is periodic. We will give some extensions of this result and raise several related questions. Read More

We obtain new inequalities for the modified Bessel function of the second kind $K_\nu$ in terms of the gamma function. These bounds follow as special cases of inequalities that we derive for the kernel of the Kr\"{a}tzel integral transformation. Read More

For the family $P:=x^n+a_1x^{n-1}+\cdots +a_n$ of complex polynomials in the variable $x$ we study its {\em discriminant} $R:=$Res$(P,P',x)$, $R\in \mathbb{C}[a]$, $a=(a_1,\ldots ,a_n)$. When $R$ is regarded as a polynomial in $a_k$, one can consider its discriminant $\tilde{D}_k:=$Res$(R,\partial R/\partial a_k,a_k)$. We show that $\tilde{D}_k=c_k(a_n)^{d(n,k)}M_k^2T_k^3$, where $c_k\in \mathbb{Q}^*$, $d(n,k):=\min (1,n-k)+\max (0,n-k-2)$, the polynomials $M_k,T_k\in \mathbb{C}[a^k]$ have integer coefficients, $a^k=(a_1,\ldots ,a_{k-1},a_{k+1},\ldots ,a_n)$, the sets $\{ M_k=0\}$ and $\{ T_k=0\}$ are the projections in the space of the variables $a^k$ of the closures of the strata of the variety $\{ R=0\}$ on which $P$ has respectively two double roots or a triple root. Read More

Let $T\subset{\mathbb R}^n$ be a fixed set. By a scaled copy of $T$ around $x\in{\mathbb R}^n$ we mean a set of the form $x+rT$ for some $r>0$. In this survey paper we study results about the following type of problems: How small can a set be if it contains a scaled copy of $T$ around every point of a set of given size? We will consider the cases when $T$ is circle or sphere centered at the origin, Cantor set in ${\mathbb R}$, the boundary of a square centered at the origin, or more generally the $k$-skeleton ($0\le kRead More

This is Chapter 2 for Volume 2, entitled {\it Multivariable special functions}, of the Askey-Bateman project. We provide an overview of the basic results on orthogonal polynomials of several variables. After a short introduction on general properties, the main text concentrates on specific systems of orthogonal polynomials in two and more variables that correspond to, or are generalizations of, classical orthogonal polynomials of one variable. Read More

We prove that bilinear fractional integral operators and similar multipliers are smoothing in the sense that they improve the regularity of functions. We also treat bilinear singular multiplier operators which preserve regularity and obtain several Leibniz-type rules in the contexts of Lebesgue and mixed Lebesgue spaces. Read More

A numerical procedure and its MAPLE implementation capable of rigorously, albeit in a brute-force manner, proving specific strict one-variable inequalities in specific finite intervals is described. The procedure is useful, for instance, to affirm strict lower bounds of specific functions. Read More

We determine the asymptotic behavior of the $l_{p}$-norms of the sequence of Taylor coefficients of $b^{n}$, where $b=\frac{z-\lambda}{1-\bar{\lambda}z}$ is an automorphism of the unit disk, $p\in[1,\infty]$, and $n$ is large. It is known that in the parameter range $p\in[1,2]$ a sharp upper bound \begin{align*} \left|\!\left|b^{n}\right|\!\right|_{l_{p}^A}\leq C_{p}n^{\frac{2-p}{2p}} \end{align*} holds. In this article we find that this estimate is valid even when $p\in[1,4)$. Read More

We investigate mixed Lusin area integrals associated with Jacobi trigonometric polynomial expansions. We prove that these operators can be viewed as vector-valued Calder\'on-Zygmund operators in the sense of the associated space of homogeneous type. Consequently, their various mapping properties, in particular on weighted $L^p$ spaces, follow from the general theory. Read More

We introduce a new definition of topological degree for a meaningful class of operators which need not be continuous. Subsequently, we derive a number of fixed point theorems for such operators. As an application, we deduce a new existence result for first-order ODEs with discontinuous nonlinearities. Read More

In this work study the associated Legendre functions of the second kind with a purely imaginary argument $Q^k_\ell(i x)$. We derive the conditions under which they provide a set of square integrable functions when $x \in \mathbb{R}$ and we prove the relevant orthogonality relation that they satisfy. Read More

The Darlington synthesis problem (in the scalar case) is the problem of embedding a given contractive analytic function to an inner $2\times 2$ matrix function as the entry. A fundamental result of Arov--Douglas--Helton relates this algebraic property to a pure analytic one known as a pseudocontinuation of bounded type. We suggest a local version of the Darlington synthesis problem and prove a local analog of the ADH theorem. Read More

We introduce the so called convex body valued sparse operators, which generalize the notion of sparse operators to the case of spaces of vector valued functions. We prove that Calder\'on--Zygmund operators as well as Haar shifts and paraproducts can be dominated by such operators. By estimating sparse operators we obtain weighted estimates with matrix weights. Read More

We consider a signal reconstruction problem for signals $F$ of the form $ F(x)=\sum_{j=1}^{d}a_{j}\delta\left(x-x_{j}\right),$ from their moments $m_k(F)=\int x^kF(x)dx.$ We assume $m_k(F)$ to be known for $k=0,1,\ldots,N,$ with an absolute error not exceeding $\epsilon > 0$. We study the "geometry of error amplification" in reconstruction of $F$ from $m_k(F),$ in situations where two neighboring nodes $x_i$ and $x_{i+1}$ near-collide, i. Read More

Our aim is to find the minimal Hausdorff dimension of the union of scaled and/or rotated copies of the $k$-skeleton of a fixed polytope centered at the points of a given set. For many of these problems, we show that a typical arrangement in the sense of Baire category gives minimal Hausdorff dimension. In particular, this proves a conjecture of R. Read More

We establish the equality among the Bernstein numbers, Isomorphism numbers and the Mityagin numbers for the Volterra operator considered between the spaces L^1([0; 1]) and C([0; 1]). We prove that each of them equals to 1/2n-1 . Moreover, we obtain that the Approximation numbers, Kolmogorov numbers and the Gelfand numbers of the Volterra operator coincide and they are equal to 1/2 when n>1 (and 1 for n = 1). Read More

Given a Muckenhoupt weight $w$ and a second order divergence form elliptic operator $L$, we consider different versions of the weighted Hardy space $H^1_L(w)$ defined by conical square functions and non-tangential maximal functions associated with the heat and Poisson semigroups generated by $L$. We show that all of them are isomorphic and also that $H^1_L(w)$ admits a molecular characterization. One of the advantages of our methods is that our assumptions extend naturally the unweighted theory developed by S. Read More

In this paper, we characterize all the irreducible Darboux polynomials and polynomial first integrals of FitzHugh-Nagumo (F-N) system. The method of the weight homogeneous polynomials and the characteristic curves is widely used to give a complete classification of Darboux polynomials of a system. However, this method does not work for F-N system. Read More

It is shown that the integrals of the Jacobi polynomials \begin{equation*}%\label{eq:Fn^J} \int_0^t (t-\theta)^\delta P_n^{(\alpha-\frac12,\beta-\frac12)}(\cos \theta) \left(\sin \tfrac{\theta}2\right)^{2 \alpha} \left(\cos \tfrac{\theta}2\right)^{2 \beta} d\theta > 0 \end{equation*} for all $t \in (0,\pi]$ and $n \in \mathbb{N}$ if $\delta \ge \alpha + 1$ for $\alpha,\beta \in \mathbb{N}_0$ and $\max\{\alpha,\beta\} > 0$. This proves a conjecture on the integral of the Gegenbauer polynomials in \cite{BCX} that implies the strictly positive definiteness of the function $\theta \mapsto (t - \theta)_+^\delta$ on the unit sphere $\mathbb{S}^{d-1}$ for $\delta \ge \lceil \frac{d}{2}\rceil$ and the Poly\`a criterion for positive definite functions on the sphere for all dimensions. Moreover, the positive definiteness of the function $\theta \mapsto (t - \theta)_+^\delta$ is also established on the compact two-point homogeneous spaces. Read More

In this paper, we discussed the non-local derivative on the fractal Cantor set. The scaling properties are given for both local and non-local fractal derivatives. The local and non-local fractal differential equations are solved and compared and related physical models are suggested. Read More

We give a simple and more elementary proof that the notions of Domain of Holomorphy and Weak Domain of Holomorphy are equivalent. This proof is based on a combination of Baire's Category Theorey and Montel's Theorem. We also obtain generalizations by demanding that the non-extentable functions belong to a particular class of holomorphic functions in the domain. Read More

In this paper we introduce the Two Parameter Gamma Function, Beta Function and Pochhammer Symbol. We named them, as p - k Gamma Function, p - k Beta Function and p - k Pochhammer Symbol and denoted as $_{p}\Gamma_{k}(x), $ $_{p}B_{k}(x,y) $ and $_{p}(x)_{n,k} $ respectively. We prove the several identities for $_{p}\Gamma_{k}(x), $ $_{p}B_{k}(x,y) $ and $_{p}(x)_{n,k} $ those satisfied by the classical Gamma, Beta and Pochhammer Symbol. Read More

The discrete Heisenberg group $\mathbb{H}_{\mathbb{Z}}^{2k+1}$ is the group generated by $a_1,b_1,\ldots,a_k,b_k,c$, subject to the relations $[a_1,b_1]=\ldots=[a_k,b_k]=c$ and $[a_i,a_j]=[b_i,b_j]=[a_i,b_j]=[a_i,c]=[b_i,c]=1$ for every distinct $i,j\in \{1,\ldots,k\}$. Denote $S=\{a_1^{\pm 1},b_1^{\pm 1},\ldots,a_k^{\pm 1},b_k^{\pm 1}\}$. The horizontal boundary of $\Omega\subset \mathbb{H}_{\mathbb{Z}}^{2k+1}$, denoted $\partial_{h}\Omega$, is the set of all $(x,y)\in \Omega\times (\mathbb{H}_{\mathbb{Z}}^{2k+1}\setminus \Omega)$ such that $x^{-1}y\in S$. Read More

Conformable fractional derivative is introduced by the authors Khalil et al. In this study we develop their concept and introduce multivariable conformable derivative for a vector valued function with several variables. Read More

Let $\phi$ be a smooth function on a compact interval $I$. Let $$\gamma(t)=\left (t,t^2,\cdots,t^{n-1},\phi(t)\right).$$ In this paper, we show that $$\left(\int_I \big|\hat f(\gamma(t))\big|^q \big|\phi^{(n)}(t)\big|^{\frac{2}{n(n+1)}} dt\right)^{1/q}\le C\|f\|_{L^p(\mathbb R^n)}$$ holds in the range $$1\le p<\frac{n^2+n+2}{n^2+n},\quad 1\le q<\frac{2}{n^2+n}p'. Read More

Hysteresis operators appear in many applications such as elasto-plasticity and micromagnetics, and can be used for a wider class of systems, where rate-independent memory plays a role. A natural approximation for hysteresis operators are fast-slow dynamical systems, which - in their used approximation form - do not involve any memory effects. Hence, viewing hysteresis operators as a limit of approximating fast-slow dynamics involves subtle limit procedures. Read More

Under a mild Lipschitz condition we prove a theorem on the existence and uniqueness of global solutions to delay fractional differential equations. Then, we establish a result on the exponential boundedness for these solutions. Read More

It is known, that among the formal solutions of the sixth Painlev\'e equation there met series with integer power exponents of the independent variable $x$ with coefficients in form of formal Laurent series (with finite main parts) in $\log^{-1} x$ (complicated expansions), or in $x^{{\rm i}\,\theta}$, where ${\rm i}=\sqrt{-1},$ $\theta\in\mathbb{R},$ $\theta\neq 0$ (exotic expansions). These coefficients can be computed consecutively. Here we research analytic properties of the series, that are the coefficients of the complicated and exotic formal solutions of the sixth Painlev\'e equation. Read More

We develop the viability theorem for the mean field type control system. It is a dynamical system in the Wasserstein space describing an evolution of a large population of agents in mean-field interaction under a control of a unique decision maker. We introduce a tangent distribution to a subset of the Wasserstein space. Read More

The subject of fractional calculus has witnessed rapid development over past few decades. In particular the area of fractional differential equations has received considerable attention. Several theoretical results have been obtained and powerful numerical methods have been developed. Read More