On the number of distinct prime factors of nj+a^hk

Let \(\omega (n)\) denote the number of distinct prime factors of \(n\) . Then for any given \(K\ge 2\) , small \(\epsilon >0\) and sufficiently large (only depending on \(K\) and \(\epsilon \) ) \(x\) , there exist at least \(x^{1-\epsilon }\) integers \(n\in [x,(1+K^{-1})x]\) such that \(\omega (nj\pm a^hk)\ge (\log \log \log x)^{\frac{1}{3}-\epsilon }\) for all \(2\le a\le K\) , \(1\le j,k\le K\) and \(0\le h\le K\log x\) .     

Refined methods for the identifiability of tensors

We prove that the general tensor of size \(2^n\) and rank \(k\) has a unique decomposition as the sum of decomposable tensors if \(k\le 0.9997\frac{2^n}{n+1}\) (the constant 1 being the optimal value). Similarly, the general tensor of size \(3^n\) and rank \(k\) has a unique decomposition as the sum of decomposable tensors if \(k\le 0.998\frac{3^n}{2n+1}\) (the constant 1 being the optimal value). Some results of this flavor are obtained for tensors of any size, but the explicit bounds obtained are weaker.     

Regularity and uniqueness of a class of biharmonic map heat flows

We consider a class of weak solutions of the heat flow of biharmonic maps from \(\Omega \subset \mathbb{R }^n\) to the unit sphere \(\mathbb{S }^L\subset \mathbb{R }^{L+1}\) , that have small renormalized total energies locally at each interior point. For any such a weak solution, we prove the interior smoothness, and the properties of uniqueness, convexity of hessian energy, and unique limit at \(t=\infty \) . We verify that any weak solution \(u\) to the heat flow of biharmonic maps from \(\Omega \) to a compact Riemannian manifold \(N\) without boundary, with \(\nabla ^2 u\in L^q_tL^p_x\) for some \(p>\frac{n}{2}\) and \(q>2\) satisfying (1.12), has small renormalized total energy locally and hence enjoys both the interior smoothness and uniqueness property. Finally, if an initial data \(u_0\in W^{2,r}(\mathbb{R }^n, N)\) for some \(r>\frac{n}{2}\) , then we establish the local existence of heat flow of biharmonic maps \(u\) , with \(\nabla ^2 u\in L^q_tL^p_x\) for some \(p>\frac{n}{2}\) and \(q>2\) satisfying (1.12).     

Distributions and the Global Operator of Slice Monogenic Functions

In this paper we consider functions \(f\) defined on an open set \(U\) of the Euclidean space \(\mathbb{R }^{n+1}\) and with values in the Clifford Algebra \(\mathbb{R }_n\) . Slice monogenic functions \(f: U \subseteq \mathbb{R }^{n+1} \rightarrow \mathbb{R }_n\) belong to the kernel of the global differential operator with non constant coefficients given by \( \mathcal{G }=|{\underline{x}}|^2\frac{\partial }{\partial x_0} \ + \ {\underline{x}} \ \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}. \) Since the operator \(\mathcal{G }\) is not elliptic and there is a degeneracy in \( {\underline{x}}=0\) , its kernel contains also less smooth functions that have to be interpreted as distributions. We study the distributional solutions of the differential equation \(\mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})\) and some of its variations. In particular, we focus our attention on the solutions of the differential equation \( ({\underline{x}}\frac{\partial }{\partial x_0} \ - E)F(x_0,{\underline{x}})=G(x_0,{\underline{x}}), \) where \(E= \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}\) is the Euler operator, from which we deduce properties of the solutions of the equation \( \mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})\) .     

On cycles for the doubling map which are disjoint from an interval

Let \(T:[0,1]\rightarrow [0,1]\) be the doubling map and let \(0 . We say that an integer \(n\ge 3\) is bad for \((a,b)\) if all \(n\) -cycles for \(T\) intersect \((a,b)\) . Let \(B(a,b)\) denote the set of all \(n\) which are bad for \((a,b)\) . In this paper we completely describe the sets: $$\begin{aligned} D_2=\{(a,b) : B(a,b)\,\text {is finite}\} \end{aligned}$$ and $$\begin{aligned} D_3=\{(a,b) : B(a,b)=\varnothing \}. \end{aligned}$$ In particular, we show that if \(b-a<\frac{1}{6}\) , then \((a,b)\in D_2\) , and if \(b-a\le \frac{2}{15}\) , then \((a,b)\in D_3\) , both constants being sharp.     

Randomized large distortion dimension reduction

Consider a random matrix \(H:{\mathbb {R}}^{n}\longrightarrow {\mathbb {R}}^{m}\) . Let \(D\ge 2\) and let \(\{W_l\}_{l=1}^{p}\) be a set of \(k\) -dimensional affine subspaces of \({\mathbb {R}}^{n}\) . We ask what is the probability that for all \(1\le l\le p\) and \(x,y\in W_l\) , $$\begin{aligned} \Vert x-y\Vert _2\le \Vert Hx-Hy\Vert _2\le D\Vert x-y\Vert _2. \end{aligned}$$ We show that for \(m=O\big (k+\frac{\ln {p}}{\ln {D}}\big )\) and a variety of different classes of random matrices \(H\) , which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on \(m\) is tight in terms of \(k,p,D\) .     

Small secret exponent attack on RSA variant with modulus N=p^rq

We consider an RSA variant with Modulus \(N=p^rq\) . This variant is known as Prime Power RSA. In PKC 2004, May proved when decryption exponent \(d<N^{ \frac{r}{(r+1)^2}}\) or \(d< N^{\left( \frac{r-1}{r+1}\right) ^2}\) , one can factor \(N\) in polynomial time. In this paper, we improve this bound when \(r \le 5\) . We provide detailed experimental results to justify our claim.     

Additive mappings satisfying algebraic conditions in rings

Let \(R\) be any \((n+1)!\) -torsion free ring and \(F,D: R\rightarrow R\) be additive mappings satisfying \(F(x^{n+1})=(\alpha (x))^nF(x)+\sum \nolimits _{i=1}^n (\alpha (x))^{n-i}(\beta (x))^iD(x)\) for all \(x\in R\) , where \(n\) is a fixed integer and \(\alpha \) , \(\beta \) are automorphisms of \(R\) . Then, \(D\) is Jordan left \((\alpha , \beta )\) -derivation and \(F\) is generalized Jordan left \((\alpha , \beta )\) -derivation on \(R\) and if additive mappings \(F\) and \(D\) satisfying \(F(x^{n+1})=F(x)(\alpha (x))^n+\sum \nolimits _{i=1}^n (\beta (x))^iD(x)(\alpha (x))^{n-i}\) for all \(x\in R\) . Then, \(D\) is Jordan \((\alpha , \beta )\) -derivation and \(F\) is generalized Jordan \((\alpha , \beta )\) -derivation on \(R\) . At last some immediate consequences of the above theorems have been given.     

Two families of approximations for the gamma function

In this paper, we establish two families of approximations for the gamma function: $$ \begin{array}{lll} {\varGamma}(x+1)&=\sqrt{2\pi x}{\left({\frac{x+a}{{\mathrm{e}}}}\right)}^x {\left({\frac{x+a}{x-a}}\right)}^{-\frac{x}{2}+\frac{1}{4}} {\left({\frac{x+b}{x-b}}\right)}^{\sum\limits_{k=0}^m\frac{{\beta}_k}{x^{2k}}+O{{\left(\frac{1}{x^{2m+2}}\right)}}},\\ {\varGamma}(x+1)&=\sqrt{2\pi x}\cdot(x+a)^{\frac{x}{2}+\frac{1}{4}}(x-a)^{\frac{x}{2}-\frac{1}{4}} {\left({\frac{x-1}{x+1}}\right)}^{\frac{x^2}{2}}\\ &\quad\times {\left({\frac{x-c}{x+c}}\right)}^{\sum\limits_{k=0}^m\frac{{\gamma}_k}{x^{2k}}+O{\left({\frac{1}{x^{2m+2}}}\right)}}, \end{array}$$ where the constants ${\beta }_k$ and ${\gamma }_k$ can be determined by recurrences, and $a$ , $b$ , $c$ are parameters. Numerical comparison shows that our results are more accurate than Stieltjes, Luschny and Nemes’ formulae, which, to our knowledge, are better than other approximations in the literature.     

On Sharp Aperture-Weighted Estimates for Square Functions

Let \(S_{\alpha ,\psi }(f)\) be the square function defined by means of the cone in \({\mathbb R}^{n+1}_{+}\) of aperture \(\alpha \) , and a standard kernel \(\psi \) . Let \([w]_{A_p}\) denote the \(A_p\) characteristic of the weight \(w\) . We show that for any \(1<p<\infty \) and \(\alpha \ge 1\) , $$\begin{aligned} \Vert S_{\alpha ,\psi }\Vert _{L^p(w)}\lesssim \alpha ^n[w]_{A_p}^{\max \left( \frac{1}{2},\frac{1}{p-1}\right) }. \end{aligned}$$ For each fixed \(\alpha \) the dependence on \([w]_{A_p}\) is sharp. Also, on all class \(A_p\) the result is sharp in \(\alpha \) . Previously this estimate was proved in the case \(\alpha =1\) using the intrinsic square function. However, that approach does not allow to get the above estimate with sharp dependence on \(\alpha \) . Hence we give a different proof suitable for all \(\alpha \ge 1\) and avoiding the notion of the intrinsic square function.     

The Hardy–Rellich Inequality and Uncertainty Principle on the Sphere

Let \(\Delta _0\) be the Laplace–Beltrami operator on the unit sphere \(\mathbb {S}^{d-1}\) of \({\mathbb R}^d\) . We show that the Hardy–Rellich inequality of the form $$\begin{aligned} \mathop \int \limits _{\mathbb {S}^{d-1}} \left| f (x)\right| ^2 \mathrm{d}{\sigma }(x) \le c_d \min _{e\in \mathbb {S}^{d-1}} \mathop \int \limits _{\mathbb {S}^{d-1}} (1- {\langle }x, e {\rangle }) \left| (-\Delta _0)^{\frac{1}{2}}f(x) \right| ^2 \mathrm{d}{\sigma }(x) \end{aligned}$$ holds for \(d =2\) and \(d \ge 4\) but does not hold for \(d=3\) with any finite constant, and the optimal constant for the inequality is \(c_d = 8/(d-3)^2\) for \(d =2, 4, 5,\) and, under additional restrictions on the function space, for \(d\ge 6\) . This inequality yields an uncertainty principle of the form $$\begin{aligned} \min _{e\in \mathbb {S}^{d-1}} \mathop \int \limits _{\mathbb {S}^{d-1}} (1- {\langle }x, e {\rangle }) |f(x)|^2 \mathrm{d}{\sigma }(x) \mathop \int \limits _{\mathbb {S}^{d-1}}\left| \nabla _0 f(x)\right| ^2 \mathrm{d}{\sigma }(x) \ge c'_d \end{aligned}$$ on the sphere for functions with zero mean and unit norm, which can be used to establish another uncertainty principle without zero mean assumption, both of which appear to be new.     

Limit theorems for von Mises statistics of a measure preserving transformation

For a measure preserving transformation \(T\) of a probability space \((X,\mathcal{F },\mu )\) and some \(d \ge 1\) we investigate almost sure and distributional convergence of random variables of the form $$\begin{aligned} x \rightarrow \frac{1}{C_n} \sum _{0\le i_1,\ldots ,\,i_d where \(C_1, C_2,\ldots \) are normalizing constants and the kernel \(f\) belongs to an appropriate subspace in some \(L_p(X^d\!,\, \mathcal{F }^{\otimes d}\!,\,\mu ^d)\) . We establish a form of the individual ergodic theorem for such sequences. Using a filtration compatible with \(T\) and the martingale approximation, we prove a central limit theorem in the non-degenerate case; for a class of canonical (totally degenerate) kernels and \(d=2\) , we also show that the convergence holds in distribution towards a quadratic form \(\sum _{m=1}^{\infty } \lambda _m\eta ^2_m\) in independent standard Gaussian variables \(\eta _1, \eta _2, \ldots \) .     

On the Dynamics of WKB Wave Functions Whose Phase are Weak KAM Solutions of H–J Equation

In the framework of toroidal Pseudodifferential operators on the flat torus \({\mathbb {T}}^n := ({\mathbb {R}} / 2\pi {\mathbb {Z}})^n\) we begin by proving the closure under composition for the class of Weyl operators \(\mathrm {Op}^w_\hbar (b)\) with symbols \(b \in S^m (\mathbb {T}^n \times \mathbb {R}^n)\) . Subsequently, we consider \(\mathrm {Op}^w_\hbar (H)\) when \(H=\frac{1}{2} |\eta |^2 + V(x)\) where \(V \in C^\infty ({\mathbb {T}}^n)\) and we exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schrödinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner transform of the solution of the Schrödinger equation to the solution of the Liouville equation on \(\mathbb {T}^n \times {\mathbb {R}}^n\) written in the measure sense. These results are applied to the study of some WKB type wave functions in the Sobolev space \(H^{1} (\mathbb {T}^n; {\mathbb {C}})\) with phase functions in the class of Lipschitz continuous weak KAM solutions (positive and negative type) of the Hamilton–Jacobi equation \(\frac{1}{2} |P+ \nabla _x v (P,x)|^2 + V(x) = \bar{H}(P)\) for \(P \in \ell {\mathbb {Z}}^n\) with \(\ell >0\) , and to the study of the backward and forward time propagation of the related Wigner measures supported on the graph of \(P+ \nabla _x v\) .     

The Fourier transform of multiradial functions

We obtain an exact formula for the Fourier transform of multiradial functions, i.e., functions of the form \(\varPhi (x)=\phi (|x_1|, \dots , |x_m|), x_i\in \mathbf R^{n_i}\) , in terms of the Fourier transform of the function \(\phi \) on \(\mathbf R^{r_1}\times \cdots \times \mathbf R^{r_m}\) , where \(r_i\) is either \(1\) or \(2\) .     

Uncertainty principles and sum complexes

Let \(p\) be a prime and let \(A\) be a nonempty subset of the cyclic group \(C_p\) . For a field \({\mathbb F}\) and an element \(f\) in the group algebra \({\mathbb F}[C_p]\) let \(T_f\) be the endomorphism of \({\mathbb F}[C_p]\) given by \(T_f(g)=fg\) . The uncertainty number \(u_{{\mathbb F}}(A)\) is the minimal rank of \(T_f\) over all nonzero \(f \in {\mathbb F}[C_p]\) such that \(\mathrm{supp}(f) \subset A\) . The following topological characterization of uncertainty numbers is established. For \(1 \le k \le p\) define the sum complex \(X_{A,k}\) as the \((k-1)\) -dimensional complex on the vertex set \(C_p\) with a full \((k-2)\) -skeleton whose \((k-1)\) -faces are all \(\sigma \subset C_p\) such that \(|\sigma |=k\) and \(\prod _{x \in \sigma }x \in A\) . It is shown that if \({\mathbb F}\) is algebraically closed then $$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{k :\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$ The main ingredient in the proof is the determination of the homology groups of \(X_{A,k}\) with field coefficients. In particular it is shown that if \(|A| \le k\) then \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)      

Gaussian Integral Means of Entire Functions

For an entire function \(f:\mathbb C\mapsto \mathbb C\) and a triple \((p,\alpha , r)\in (0,\infty )\times (-\infty ,\infty )\times (0,\infty ]\) , the Gaussian integral mean of \(f\) (with respect to the area measure \(dA\) ) is defined by $$\begin{aligned} {\mathsf M}_{p,\alpha }(f,r)=\left( \,\, {\int \limits _{|z| Via deriving a maximum principle for \({\mathsf M}_{p,\alpha }(f,r)\) , we establish not only Fock–Sobolev trace inequalities associated with \({\mathsf M}_{p,p/2}(z^m f(z),\infty )\) (as \(m=0,1,2,\ldots \) ), but also convexities of \(r\mapsto \ln {\mathsf M}_{p,\alpha }(z^m,r)\) and \(r\mapsto {\mathsf M}_{2,\alpha <0}(f,r)\) in \(\ln r\) with \(0 .     

Global cauchy problem of 2D generalized MHD equations

In this paper we prove the global-in-time existence of smooth solutions of the 2D generalized MHD system with fractional diffusion \((-\Delta )^\alpha u\) , \(0<\alpha <\frac{1}{2}\) .     

Boundedness of Singular Integrals in Journé’s Class on Weighted Multiparameter Hardy Spaces

In this paper, we obtain the boundedness of singular integral operators T in Journé’s class on weighted multiparameter Hardy spaces \(H^{p}_{w}\) of arbitrary k number of parameters (k≥3) under the assumption that \(T^{\ast}_{i}(1)=0\) , i=1,…,k, and the kernel of T has a regularity of order ?>0, where \(w \in A_{r}(\Bbb{R}^{n_{1}}\times \cdots \times \Bbb{R}^{n_{k}}), r \geq 1\) and \(\max\{ \frac{r n_{1} }{n_{1}+\varepsilon}, \ldots, \frac{r n_{k} }{n_{k}+\varepsilon}\} .     

Asymptotics of Landau Constants with Optimal Error Bounds

We study the asymptotic expansion for the Landau constants \(G_n\) , $$\begin{aligned} \pi G_n\sim \ln N + \gamma +4\ln 2 + \sum _{s=1}^\infty \frac{\beta _{2s}}{ N^{2s}},\quad n\rightarrow \infty , \end{aligned}$$ where \(N=n+3/4, \gamma =0.5772\ldots \) is Euler’s constant, and \((-1)^{s+1}\beta _{2s}\) are positive rational numbers, given explicitly in an iterative manner. We show that the error due to truncation is bounded in absolute value by, and of the same sign as, the first neglected term for all nonnegative \(n\) . Consequently, we obtain optimal sharp bounds up to arbitrary orders of the form $$\begin{aligned} \ln N+\gamma +4\ln 2+\sum _{s=1}^{2m}\frac{\beta _{2s}}{N^{2s}}< \pi G_n < \ln N+\gamma +4\ln 2+\sum _{s=1}^{2k-1}\frac{\beta _{2s}}{N^{2s}} \end{aligned}$$ for all \(n=0,1,2,\ldots , m=1,2,\ldots \) , and \(k=1,2,\ldots \) . The results are proved by approximating the coefficients \(\beta _{2s}\) with the Gauss hypergeometric functions involved and by using the second-order difference equation satisfied by \(G_n\) , as well as an integral representation of the constants \(\rho _k=(-1)^{k+1}\beta _{2k}/(2k-1)!\) .     

Borderline gradient continuity for nonlinear parabolic systems

We consider the evolutionary \(p\) -Laplacean system $$\begin{aligned} \partial _t u-\triangle _p u=F,\qquad p > \frac{2n}{n+2} \end{aligned}$$ in cylindrical domains of \( \mathbb R^{n}\times \mathbb R\) , and prove the continuity of the spatial gradient \(Du\) under the Lorentz space assumption \(F\in L(n+2,1)\) . When \(F\) is time independent the condition improves in \(F \in L(n,1)\) . This is the limiting case of a result of DiBenedetto claiming that \(Du\) is Hölder continuous when \(F \in L^{q}\) for \(q>n+2\) . At the same time, this is the natural nonlinear parabolic analog of a linear result of Stein, claiming the gradient continuity of solutions to the linear elliptic system \(\triangle u \in L(n,1)\) is continuous. New potential estimates are derived and moreover suitable nonlinear potentials are used to describe fine properties of solutions.