Least zero of a cubic form

An effective search bound is established for the least non-trivial integer zero of an arbitrary cubic form \({C \in \mathbb {Z}[X_1,\ldots,X_n]}\), provided that n ≥ 17.     

On convex hull of Gaussian samples

Let X _i = {X _i (t), t ∈ T} be i.i.d. copies of a centered Gaussian process X = {X(t), t ∈ T} with values in\( {\mathbb{R}^d} \) defined on a separable metric space T. It is supposed that X is bounded. We consider the asymptotic behavior of convex hulls

$ {W_n} = {\text{conv}}\left\{ {{X_1}(t), \ldots, {X_n}(t),\,\,t \in T} \right\} $

and show that, with probability 1,

$ \mathop {{\lim }}\limits_{n \to \infty } \frac{1}{{\sqrt {{2\ln n}} }}{W_n} = W $

(in the sense of Hausdorff distance), where the limit shape W is defined by the covariance structure of X: W = conv{K _t , t ∈ T}, Kt being the concentration ellipsoid of X(t). We also study the asymptotic behavior of the mathematical expectations E f(W _n ), where f is an homogeneous functional.

     

On weighted absolute convergence of multiple Fourier series of a function of $${(p_{1} ,\ldots,p_{N})}$$–$${(\Lambda^1 ,\ldots ,\Lambda^N)}$$(Λ1,…,ΛN)– BV

We obtain a sufficient condition for the weighted β-absolute convergence of the multiple Fourier series

$$\sum_{k_{1} \in\mathbb{Z}} \cdots \sum_{k_{N} \in\mathbb{Z}} \gamma_{k_{1} ,\ldots ,k_{N}} |\hat{f}(k_1 ,\ldots ,k_N)|^\beta,$$

where \({0 < \beta < 2}\) and \({\gamma_{k_{1} ,\ldots ,k_{N}}}\) is a weight sequence, of a function f of the class \({(\Lambda^1 ,\ldots ,\Lambda^N)BV^{(p_{1} ,\ldots ,p_{N})}([0, 2\pi]^N)}\).

     

Generalized Skew Derivations Cocentralizing Multilinear Polynomials

Let R be a prime ring of characteristic different from 2, Q _r be its right Martindale quotient ring and C be its extended centroid. Suppose that F, G are generalized skew derivations of R and \({f(x_1, \ldots, x_n)}\) is a non-central multilinear polynomial over C with n non-commuting variables. If F and G satisfy the following condition:

$$F(f(r_1,\ldots, r_n))f(r_1, \ldots,r_n)-f(r_1,\ldots,r_n)G(f(r_1,\ldots, r_n))\in C$$

for all \({r_1, \ldots, r_n \in R}\), then we describe all possible forms of F and G.

     

Polynomial partitioning for several sets of varieties

We give a new, systematic proof for a recent result of Larry Guth and thus also extend the result to a setting with several families of varieties: For any integer \(D\ge 1\) and any collection of sets \(\Gamma _1,\ldots ,\Gamma _j\) of low-degree k-dimensional varieties in \(\mathbb {R}^n\), there exists a non-zero polynomial \(p\in \mathbb {R}[X_1,\ldots ,X_n]\) of degree at most D, so that each connected component of \(\mathbb {R}^n{\setminus }Z(p)\) intersects \(O(jD^{k-n}|\Gamma _i|)\) varieties of \(\Gamma _i\), simultaneously for every \(1\le i\le j\). For \(j=1\), we recover the original result by Guth. Our proof, via an index calculation in equivariant cohomology, shows how the degrees of the polynomials used for partitioning are dictated by the topology, namely, by the Euler class being given in terms of a top Dickson polynomial.     

A Refinement of the Kolmogorov–Marcinkiewicz–Zygmund Strong Law of Large Numbers

Let {X _n ; n≥1} be a sequence of independent copies of a real-valued random variable X and set S _n =X ₁+???+X _n , n≥1. This paper is devoted to a refinement of the classical Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers. We show that for 0<p<2,

$\sum_{n=1}^{\infty}\frac{1}{n}\biggl(\frac{|S_{n}|}{n^{1/p}}\biggr)<\infty\quad \mbox{almost surely}$

if and only if

$\begin{cases}\mathbb{E}|X|^{p}<\infty &; \mbox{if }0 < p < 1,\\\mathbb{E}X=0,\ \sum_{n=1}^{\infty}\frac{|\mathbb{E}XI\{|X|\leq n\}|}{n}<\infty,\mbox{ and }\\\sum_{n=1}^{\infty}\frac{\int_{\min\{u_{n},n\}}^{n}\mathbb{P}(|X|>t)\,dt}{n}<\infty &; \mbox{if }p = 1,\\\mathbb{E}X=0\mbox{ and }\int_{0}^{\infty}\mathbb{P}^{1/p}(|X|>t)\,dt<\infty,&;\mbox{if }1 < p < 2,\end{cases}$

where \(u_{n}=\inf \{t:~\mathbb{P}(|X|>t)<\frac{1}{n}\}\), n≥1. Versions of the above result in a Banach space setting are also presented. To establish these results, we invoke the remarkable Hoffmann-Jørgensen (Stud. Math. 52:159–186, 1974) inequality to obtain some general results for sums of the form \(\sum_{n=1}^{\infty}a_{n}\|\sum_{i=1}^{n}V_{i}\|\) (where {V _n ; n≥1} is a sequence of independent Banach-space-valued random variables, and a _n ≥0, n≥1), which may be of independent interest, but which we apply to \(\sum_{n=1}^{\infty}\frac{1}{n}(\frac{|S_{n}|}{n^{1/p}})\).

     

Boundedness and compactness of an integral operator in a mixed norm space on the polydisk

We study the following integral type operator

$T_g (f)(z) = \int\limits_0^{z_{} } { \cdots \int\limits_0^{z_n } {f(\zeta _1 , \ldots ,\zeta _n )} g(\zeta _1 , \ldots ,\zeta _n )d\zeta _1 , \ldots ,\zeta _n } $

in the space of analytic functions on the unit polydisk U ⁿ in the complex vector space ?ⁿ. We show that the operator is bounded in the mixed norm space

Open image in new window

, with p, q ∈ [1, ∞) and α = (α₁, …, α_n), such that α_j > ?1, for every j = 1, …, n, if and only if \(\sup _{z \in U^n } \prod\nolimits_{j = 1}^n {\left( {1 - \left| {z_j } \right|} \right)} \left| {g(z)} \right| < \infty \). Also, we prove that the operator is compact if and only if \(\lim _{z \to \partial U^n } \prod\nolimits_{j = 1}^n {\left( {1 - \left| {z_j } \right|} \right)} \left| {g(z)} \right| = 0\).

     

Derivations cocentralizing multilinear polynomials on left ideals

Let R be a prime ring with extended centroid C, λ a nonzero left ideal of R and f (X ₁, . . . , X _t ) a nonzero multilinear polynomial over C. Suppose that d and δ are derivations of R such that

$d(f(x_{1},\ldots,x_{t}))f(x_{1},\ldots,x_{t})-f(x_{1},\ldots,x_{t})\delta(f(x_{1},\ldots,x_{t}))\in C$

for all \({x_1,\ldots,x_t\in\lambda}\). Then either d = 0 and λ δ(λ) = 0 or λ C = RCe for some idempotent e in the socle of RC and one of the following holds:

1. (1)
   f (X₁, . . . , X _t ) is central-valued on eRCe;
    
2. (2)
   λ(d + δ)(λ) = 0 and f (X₁, . . . , X _t )² is central-valued on eRCe;
    
3. (3)
   char R = 2 and eRCe satisfies st ₄(X ₁, X ₂, X ₃, X ₄), the standard polynomial identity of degree 4.
    

     

Solvability of convolution type integro-differential equations on ℝ

The paper looks for the solutions of integro-differential equations of the form

$ - \frac{{d\varphi }}{{dx}} + A\varphi (x) = g(x) + B\int_\mathbb{R} {k(x - t)\lambda (t)\varphi (t)dt, x \in \mathbb{R}} $

in the class of functions which are absolutely continuous and of slow growth on ?. It is assumed that A and B are nonnegative parameters, 0 ≤ g ∈ L ₁ (?), 0 ≤ k ∈ L ₁ (?), ∫_? k(x) dx = 1 and 0 ≤ λ(x) ≤ 1 is a measurable function in ?. The equation is solved by a special factorization of the corresponding integro-differential operator in combination with appropriately modified standard methods of the theory of convolution type integral equations.

     

More on stability of almost surjective <Emphasis Type="Italic">ε</Emphasis>-isometries of Banach spaces

Let X and Y be two Banach spaces, and f: X → Y be a standard ε-isometry for some ε ≥ 0. In this paper, by using a recent theorem established by Cheng et al. (2013–2015), we show a sufficient condition guaranteeing the following sharp stability inequality of f: There is a surjective linear operator T: Y → X of norm one so that

$$\left\| {Tf(x) - x} \right\| \leqslant 2\varepsilon , for all x \in X.$$

As its application, we prove the following statements are equivalent for a standard ε-isometry f: X → Y:

1. (i)
   lim inf _t→∞ dist(ty, f(X))/|t| < 1/2, for all y ∈ S _Y ;
    
2. (ii)
   \(\tau(f)\equiv sup_{y\epsilon S_{Y}}\) lim inf _t→∞dist(ty, f(X))/|t| = 0;
    
3. (iii)
   there is a surjective linear isometry U: X → Y so that

   $$\left\| {f(x) - Ux} \right\| \leqslant 2\varepsilon , for all x \in X.$$
    

This gives an affirmative answer to a question proposed by Vestfrid (2004, 2015).     

Extremes of independent stochastic processes: a point process approach

For each n ≥ 1, let \(\{ X_{in}, \quad i \geqslant 1 \}\) be independent copies of a nonnegative continuous stochastic process X _n = (X _n (s)) _s∈S indexed by a compact metric space S. We are interested in the process of partial maxima \(\tilde M_{n}(t,s) =\max \{ X_{in}(s), 1 \leqslant i\leqslant [nt] \},\quad t\geq 0,\ s\in S,\) where the brackets [ ? ] denote the integer part. Under a regular variation condition on the sequence of processes X _n , we prove that the partial maxima process \(\tilde M_{n}\) weakly converges to a superextremal process \(\tilde M\) as \(n\to \infty \). We use a point process approach based on the convergence of empirical measures. Properties of the limit process are investigated: we characterize its finite-dimensional distributions, prove that it satisfies an homogeneous Markov property, and show in some cases that it is max-stable and self-similar. Convergence of further order statistics is also considered. We illustrate our results on the class of log-normal processes in connection with some recent results on the extremes of Gaussian processes established by Kabluchko.     

A radial symmetry and Liouville theorem for systems involving fractional Laplacian

We investigate the nonnegative solutions of the system involving the fractional Laplacian:

$$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {( - \Delta )^\alpha u_i (x) = f_i (u),} & {x \in \mathbb{R}^n , i = 1,2, \ldots ,m,} \\ \end{array} } \\ {u(x) = (u_1 (x),u_2 (x), \ldots ,u_m (x)),} \\ \end{array} } \right.$$

where 0 < α < 1, n > 2, f _i (u), 1 ≤ i ≤ m, are real-valued nonnegative functions of homogeneous degree p _i ≥ 0 and nondecreasing with respect to the independent variables u ₁, u ₂,..., u _m . By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point x ₀ if p _i = (n + 2α)/(n ? 2α) for each 1 ≤ i ≤ m; and the only nonnegative solution of this system is u ≡ 0 if 1 < p _i < (n + 2α)/(n ? 2α) for all 1 ≤ i ≤ m.

     

Intrinsic Hölder Continuity of Harmonic Functions

We consider the stochastic differential equation (SDE) of the form

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{rcl} dX^ x(t) &=& \sigma(X(t-)) dL(t) \\ X^ x(0)&=&x,\quad x\in{\mathbb{R}}^ d, \end{array}\right. \end{array} $$

where \(\sigma :{\mathbb {R}}^ d\to {\mathbb {R}}^ d\) is globally Lipschitz continuous and L={L(t):t≥0} is a Lévy process. Under this condition on σ it is well known that the above problem has a unique solution X. Let \((\mathcal {P}_{t})_{t\ge 0}\) be the Markovian semigroup associated to X defined by \(\left ({\mathcal {P}}_{t} f\right ) (x) := \mathbb {E} \left [ f(X^ x(t))\right ]\), t≥0, \(x\in {\mathbb {R}}^{d}\), \(f\in \mathcal {B}_{b}({\mathbb {R}}^{d})\). Let B be a pseudo–differential operator characterized by its symbol q. Fix \(\rho \in \mathbb {R}\). In this article we investigate under which conditions on σ, L and q there exist two constants γ>0 and C>0 such that

$$\left| B {\mathcal{P}}_{t} u \right|_{H^{\rho}_{2}} \le C \, t^{-\gamma} \,\left| u \right|_{H^{\rho}_{2}}, \quad \forall u \in {H^{\rho}_{2}}(\mathbb{R}^{d} ),\, t>0. $$

     

Searching for a Best Least Absolute Deviations Solution of an Overdetermined System of Linear Equations Motivated by Searching for a Best Least Absolute Deviations Hyperplane on the Basis of Given Data

We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations Xa=z, X∈?^m×n, m≥n, \(\mathbf{a}\in \mathbb{R}^{n}, \mathbf {z}\in\mathbb{R}^{m}\). This problem is equivalent to the problem of determining a best LAD-hyperplane x?a ^T x, x∈? ⁿ on the basis of given data \((\mathbf{x}_{i},z_{i}), \mathbf{x}_{i}= (x_{1}^{(i)},\ldots,x_{n}^{(i)})^{T}\in \mathbb{R}^{n}, z_{i}\in\mathbb{R}, i=1,\ldots,m\), whereby the minimizing functional is of the form

$F(\mathbf{a})=\|\mathbf{z}-\mathbf{Xa}\|_1=\sum_{i=1}^m|z_i-\mathbf {a}^T\mathbf{x}_i|.$

An iterative procedure is constructed as a sequence of weighted median problems, which gives the solution in finitely many steps. A criterion of optimality follows from the fact that the minimizing functional F is convex, and therefore the point a ^?∈? ⁿ is the point of a global minimum of the functional F if and only if 0∈?F(a ^?).

Motivation for the construction of the algorithm was found in a geometrically visible algorithm for determining a best LAD-plane (x,y)?αx+βy, passing through the origin of the coordinate system, on the basis of the data (x _i ,y _i ,z _i ),i=1,…,m.     

A Note on Harnack Inequalities and Propagation Sets for a Class of Hypoelliptic Operators

In this paper we are concerned with Harnack inequalities for non-negative solutions u:Ω→? to a class of second order hypoelliptic ultraparabolic partial differential equations in the form

$ \mathcal{L} u:=\sum\limits_{j=1}^m X_j^2u+X_0u-\partial_tu=0 $

where Ω is any open subset of ?^N?+?1, and the vector fields X ₁, ..., X _m and \(X_0 - \partial_t\) are invariant with respect to a suitable homogeneous Lie group. Our main goal is the following result: for any fixed (x ₀,t ₀)?∈?Ω we give a geometric sufficient condition on the compact sets \(K\subseteq {\Omega}\) for which the Harnack inequality

$ \sup\limits_{K}u\le C_K\, u(x_0,t_0) $

holds for all non-negative solutions u to the equation \(\mathcal{L} u=0\). We also compare our result with an abstract Harnack inequality from potential theory.

     

On the uniform Kreiss resolvent condition

Let B be a Banach space with norm ‖ · ‖ and identity operator I. We prove that, for a bounded linear operator T in B, the strong Kreiss resolvent condition

$\parallel (T - \lambda I)^{ - k} \parallel \leqslant \frac{M}{{(|\lambda | - 1)^k }}, |\lambda | > 1,k = 1,2, \ldots ,$

implies the uniform Kreiss resolvent condition

$\left\| {\sum\limits_{k = 0}^n {\frac{{T^k }}{{\lambda ^{k + 1} }}} } \right\| \leqslant \frac{L}{{|\lambda | - 1}}, |\lambda | > 1, n = 0,1,2, \ldots .$

We establish that an operator T satisfies the uniform Kreiss resolvent condition if and only if so does the operator T ^m for each integer m ? 2.

     

Carleson measures,BMO spaces and balayages associated to Schrdinger operators

Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators.     

Lefschetz coincidence class for several maps

The aim of this paper is to define a Lefschetz coincidence class for several maps. More specifically, for maps \({f_{1}, \ldots , f_{k} : X \rightarrow N}\) from a topological space X into a connected closed n-manifold (even nonorientable) N, a cohomological class

$$L(f_{1}, \ldots , f_{k}) \in H^{n(k-1)}(X; (f_{1}, \ldots , f_{k}) ^{\ast} (R \times \Gamma^{\ast}_{N} \times \ldots \times \Gamma^{\ast} _{N}))$$

is defined in such a way that \({L(f_{1}, \ldots , f_{k}) \neq 0}\) implies that the set of coincidences

$${\rm Coin}(f_{1}, \ldots , f_{k}) = \{x \in X\,|\,f_{1}(x) = \ldots = f_{k}(x)\}$$

is nonempty.

     

Algorithms for positive semidefinite factorization

This paper considers the problem of positive semidefinite factorization (PSD factorization), a generalization of exact nonnegative matrix factorization. Given an m-by-n nonnegative matrix X and an integer k, the PSD factorization problem consists in finding, if possible, symmetric k-by-k positive semidefinite matrices \(\{A^1,\ldots ,A^m\}\) and \(\{B^1,\ldots ,B^n\}\) such that \(X_{i,j}=\text {trace}(A^iB^j)\) for \(i=1,\ldots ,m\), and \(j=1,\ldots ,n\). PSD factorization is NP-hard. In this work, we introduce several local optimization schemes to tackle this problem: a fast projected gradient method and two algorithms based on the coordinate descent framework. The main application of PSD factorization is the computation of semidefinite extensions, that is, the representations of polyhedrons as projections of spectrahedra, for which the matrix to be factorized is the slack matrix of the polyhedron. We compare the performance of our algorithms on this class of problems. In particular, we compute the PSD extensions of size \(k=1+ \lceil \log _2(n) \rceil \) for the regular n-gons when \(n=5\), 8 and 10. We also show how to generalize our algorithms to compute the square root rank (which is the size of the factors in a PSD factorization where all factor matrices \(A^i\) and \(B^j\) have rank one) and completely PSD factorizations (which is the special case where the input matrix is symmetric and equality \(A^i=B^i\) is required for all i).