Probabilistic Density-Tapered Antenna Arrays

This paper supplements an earlier one by the authors which constructed the Dedekind completion of the ring of continuous real functions on an arbitrary frame L in terms of partial continuous real functions on L. In the present paper, we provide three alternative views of it, in terms of (i) normal semicontinuous real functions on L, (ii) the Booleanization of L (in the case of bounded real functions) and the Gleason cover of L (in the general case), and (iii) Hausdorff continuous partial real functions on L. The first is the normal completion and extends Dilworth’s classical construction to the pointfree setting. The second shows that in the bounded case, the Dedekind completion is isomorphic to the lattice of bounded continuous real functions on the Booleanization of L, and that in the non-bounded case, it is isomorphic to the lattice of continuous real functions on the Gleason cover of L. Finally, the third is the pointfree version of Anguelov’s approach in terms of interval-valued functions. Two new classes of frames, cb-frames and weak cb-frames, emerge naturally in the first two representations. We show that they are conservative generalizations of their classical counterparts.

Probability measures on a real complete linear metric space $E$ are studied via their Fourier transform on $E’$ provided $E$ has the approximation property and possesses a real positive definite continuous function $\Phi (x)$ such that $||x|| > \epsilon$ implies $\Phi (0) - \Phi (x) > c(\epsilon )$ where $c(\epsilon ) > 0$. In this setting we obtain conditions on the Fourier transforms of a family of tight Borel probabilities which yield tightness of the family of measures. This then is applied to obtain necessary and sufficient conditions for a complex valued function on $E’$ to be the Fourier transform of a tight Borel probability on $E$. An extension of the Levy continuity theorem as given by ${\text {L}}$. Gross for a separable Hilbert space is obtained for such metric spaces. We also prove that various Orlicz-type spaces are in the class of spaces to which our results apply. Finally we apply our results to certain Orlicz-type sequence spaces and obtain conditions sufficient for tightness of a family of probability measures in terms of uniform convergence of the Fourier transforms on large subsets of the dual. We also obtain a more explicit form of Bochner’s theorem for these sequence spaces. The class of sequence spaces studied contains the ${l_p}$ spaces $(0 < p \leqslant 2)$ and hence these results apply to separable Hilbert space.

The array factors and the statistical properties for two types of random arrays, namely, the totally random and the binned random arrays, are compared. In totally random arrays, the array elements are distributed independently across a common aperture according to some common probability distribution function. In binned arrays, the aperture is divided into nonoverlapping bins of equal length and array elements are distributed independently, one per bin, according to some probability distribution across each of the bins. Significant differences exist in the resulting array factors and underlying statistical properties for the two types of random arrays. These differences are delineated, with particular emphasis upon the depressed near-in sidelobe behavior that is observed for the binned arrays. Tables and plots that illustrate the differences are included, and several issues pertaining to implementation are noted. >

Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems

On the Kolmogorov complexity of continuous real functions

In this paper, we investigate the existence and regularity of positive solutions for certain singular problems that involve an anisotropic ( p , q ) -Laplacian-type operator and a singular term with a variable exponent, under zero Dirichlet boundary conditions on ∂ Ω . The main equation we analyze is − ∑ i = 1 N ∂ i ( | ∂ i u ( x ) | p i − 2 ∂ i u ( x ) ) − ∑ i = 1 N ∂ i ( | ∂ i u ( x ) | q i − 2 ∂ i u ( x ) ) = f ( x ) u ( x ) γ ( x ) in Ω , where Ω is a bounded, regular domain in R N , f is a positive function belonging to a specific Lebesgue space, and γ ( x ) is a positive continuous function on Ω ¯ . In our study, we do not make comparisons between p i and q i , and as a result, we show that the solution belongs to either W 0 1 , p → ( Ω ) ∩ W 0 1 , q → ( Ω ) or W l o c 1 , p → ( Ω ) ∩ W l o c 1 , q → ( Ω ) depending on the summability of f ( x ) and the values of γ ( x ) . The results are achieved using approximation techniques that include truncation, comparison, and variational methods.

Comonotone approximation and interpolation by entire functions

In this paper, we study the existence of positivesolution for the following p-Laplacain type equations with critical nonlinearity \begin{equation*} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -\Delta_p u + V （x)|u|^{p-2}u = K(x)f(u)+P(x)|u|^{p^*-2}u, \quad x\in\mathbb{R}^N,\\ u \in \mathcal{D}^{1,p}(\mathbb{R}^N), \end{array} \right. \end{equation*}where $\Delta_p u = div(|\nabla u|^{p-2} \nabla u),\ 1 < p < N,\ p^* =\frac{Np}{N-p}$, $V(x)$, $K(x)$ are positive continuous functions which vanish atinfinity, $f$ is a function with a subcritical growth, and $P(x)$ is a bounded,nonnegative continuous function.By working in the weighted Sobolev spaces, and using variational method, weprove that the given problem has at least one positive solution.

Let $\Omega$ be a domain in $\mathbb R^N$, where $N \ge 2$ and $\partial\Omega$ is not necessarily bounded. We consider two fast diffusion equations $\partial_t u= \mbox{div}(|\nabla u|^{p-2}{\nabla u})$ and $\partial_t u= \Delta u^{m}$, where $1<p<2$ and $0<m<1$. Let $u=u(x,t)$ be the solution of either the initial-boundary value problem over $\Omega$, where the initial value equals zero and the boundary value is a positive continuous function, or the Cauchy problem where the initial datum equals a nonnegative continuous function multiplied by the characteristic function of the set $\mathbb R^N\setminus \Omega$. Choose an open ball $B$ in $\Omega$ whose closure intersects $\partial\Omega$ only at one point, and let $\alpha > \frac {(N+1)(2-p)}{2p}$ or $\alpha > \frac {(N+1)(1-m)}{4}$. Then, we derive asymptotic estimates for the integral of $u^\alpha$ over $B$ for short times in terms of principal curvatures of $\partial\Omega$ at the point, which tells us about the interaction between fast diffusion and geometry of domain.

We consider the existence of solutions of the following weighted problem: \begin{equation*} \left\{ \begin{array}{rclll}\!\!\! L_{(\sigma,\xi)}u &amp;= &amp; f(x,u)&amp; \mbox{in} \ B \\ u &amp;&gt;&amp;0 &amp;\mbox{in }B\\ u&amp;=&amp;0 &amp;\mbox{on } \partial B, \end{array} \right. \end{equation*} where \[L_{(\sigma,\xi)}u:=\!g\Bigl(\int_{B}\bigl(\sigma(x)|\nabla u|^{N}\!+\!\xi(x)|u|^{N}\bigr)dx\!\Bigr)\!\bigl[-\textmd{div} \bigl(\sigma(x)|\nabla u|^{N-2} \nabla u\bigr)\!+\!\xi(x)u^{N-1}\bigr], \] B is the unit ball of $\mathbb{R}^{N}$, $ N&gt;2$, $\sigma(x)=(\log(\frac{e}{|x|}))^{N-1}$ the singular logarithm weight with the limiting exponent $N-1$ in the Trudinger--Moser embedding, $\xi(x)$ is a positive continuous function. The Kirchoff function $g$ is positive and continuous on $(0, +\infty)$. The nonlinearities are critical or subcritical growth in view of Trudinger--Moser inequalities of double exponential type. We prove the existence of positive solution by using Mountain Pass Theorem. In the critical case, the function of Euler--Lagrange does not fulfil the requirements of Palais--Smale conditions at all levels. We dodge this problem by using adapted test functions to identify this level of compactness. multiples of a given positive divisor with real coefficients.

We prove results on exact asymptotics as for the means and probabilities , where , , is an -dimensional Wiener process, is a positive continuous function (potential) satisfying certain conditions, , and are prescribed vectors. The results are obtained by a new method developed in this paper, the Laplace method for the occupation time of a multi-dimensional Wiener process. We consider examples of monomial and radial potentials and prove results on exact asymptotics of small deviations for the probabilities as with a fixed .

Radial continuous rotation invariant valuations on star bodies

A nonoscillation theorem for sublinear Emden–Fowler equations

In this paper we study the existence of infinitely many solutions for the second-order differential equation (φp(u′(t)))′−a(t)φq(u(t))+b(t)φr(u(t))=0, t∈(0,T). coupled with Dirichlet’s boundary conditions u (0) = u(T) = 0. Here φ(t) = |t|p-2t < q < p, 1 < r < p and a,b are positive continuous functions on [0, T]. An existence theorem is proved using the variational method and the generalized Clark’s theorem due to Liu and Wang.

Consider a spectrally positive Lévy process Z with log-Laplace exponent Ψ and a positive continuous function R on (0,∞). We investigate the entrance from infinity of the process X obtained by changing time in Z with the inverse of the additive functional η(t)=∫0tdsR(Z s). We provide a necessary and sufficient condition for infinity to be an entrance boundary of the process X. Under this condition, the process can start from infinity and we study its speed of coming down from infinity. When the Lévy process has a negative drift δ:=−γ<0, sufficient conditions over R and Ψ are found for the process to come down from infinity along the deterministic function (xt,t≥0) solution to dxt=−γR(xt)dt with x0=∞. If Ψ(λ)∼cλα as λ→0, α∈(1,2], c>0 and R is regularly varying at ∞ with index θ>α, the process comes down from infinity and we find a renormalisation in law of its running infimum at small times.