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In this paper, a nonlinear version of the Extrapolation Theorem is proved and, as a corollary, a nonlinear version of Grothendieck's Theorem is presented. Finally, we prove that if $ T:X\to H$ being a pointed metric space and $ T(0)=0$ $ T^\char93 \vert _{H^*}$-summing $ (1\le q<\infty)$ is Lipschitz 1-summing. [ABSTRACT FROM AUTHOR]

In this paper, we use the Perron method to prove the existence of viscosity solutions with asymptotic behavior at infinity to Hessian equations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove that there exist measures $ d\mu(x)=\gamma(x)dx$ $ \gamma(x)=\gamma_{0}(\vert x\vert)$ $ \gamma_{0}$ $ \mathcal{M}_{\mu}$ does not map $ L^{p}_{\mu}(\mathbb{R}^{n})$ $ L^{p}_{\mu}(\mathbb{R}^{n})$. This result answers an open question of P. Sjögren and F. Soria. [ABSTRACT FROM AUTHOR]

A theory of allocation maps has been developed by J. F. Feinstein and M. J. Heath in order to prove a theorem, using Zorn's lemma, concerning the compact plane sets known as Swiss cheese sets. These sets are important since, as domains, they provide a good source of examples in the theory of uniform algebras and rational approximation. In this paper we take a more direct approach when proving their theorem by using transfinite induction and cardinality. An explicit reference to a theory of allocation maps is no longer required. Instead we find that the repeated application of a single operation developed from the final step of the proof by Feinstein and Heath is enough. [ABSTRACT FROM AUTHOR]

This paper is concerned with the exact controllability and exact observability of linear systems in the Banach space setting. It is proved that both the admissibility of control operators and the admissibility of observation operators are invariant to cross perturbations of the generator of a $ C_0$ [ABSTRACT FROM AUTHOR]

This paper studies almost periodic/automorphic monotone systems with positive translation invariance via skew-product flows. It is proved that every bounded solution of such systems is asymptotically almost periodic/automorphic. Applications are made to a chemical reaction network, especially to enzymatic futile cycles with almost periodic/automorphic reaction coefficients. [ABSTRACT FROM AUTHOR]

Let $ \nabla f$ $ (w_1,\dotsc,w_n)$. In this paper, we prove that the Łojasiewicz Exponent $ \theta$ is precisely equal to $ \displaystyle{\max_{0\leq i\leq n}}w_i-1$, $ \vert\nabla f(z)\vert\geq c\vert z\vert^\theta$ which provides the optimal lower estimate of $ \vert\nabla f(z)\vert$ [ABSTRACT FROM AUTHOR]

It is well known that the average position or barycenter of generic orbits for the standard tent map is $ 0.5$. In this paper we prove that for any positive integer $ n$ distinct periodic orbits for the standard tent map with the same barycenter. We also provide some patterns of periodic orbits with the same barycenter. [ABSTRACT FROM AUTHOR]

Let $ u$-Laplacian type ($ 1

The aim of this paper is to prove a stronger version of a conjecture posed earlier on the existence of nondominated scalar-valued $m$-homogeneous polynomials, $m geq 3$, on arbitrary infinite dimensional Banach spaces. [ABSTRACT FROM AUTHOR]

Jacobi proved an elegant identity involving eight-fold infinite products. In this paper, we give a new proof of this identity. A key ingredient of our proof is an identity satisfied by a Ramanujan-Selberg continued fraction. [ABSTRACT FROM AUTHOR]

We are interested in the large-time behavior of periodic entropy solutions in $L^infty $ to anisotropic degenerate parabolic-hyperbolic equations of second order. Unlike the pure hyperbolic case, the nonlinear equation is no longer self-similar invariant, and the diffusion term in the equation significantly affects the large-time behavior of solutions; thus the approach developed earlier, based on the self-similar scaling, does not directly apply. In this paper, we develop another approach for establishing the decay of periodic solutions for anisotropic degenerate parabolic-hyperbolic equations. The proof is based on the kinetic formulation of entropy solutions. It involves time translations and a monotonicity-in-time property of entropy solutions and employs the advantages of the precise kinetic equation for the solutions in order to recognize the role of nonlinearity-diffusivity of the equation. [ABSTRACT FROM AUTHOR]

Let $N(a,m)$ be the least integer $n$ (if it exists) such that $varphi (n)equiv apmod m$. Friedlander and Shparlinski proved that for any $varepsilon >0$ there exists $A=A(varepsilon )>0$ such that for any positive integer $m$ which has no prime divisors $p<(log m)^A$ and any integer $a$ with $gcd (a,m)=1,$ we have the bound $N(a,m)ll m^{3+varepsilon }.$ In the present paper we improve this bound to $N(a,m)ll m^{2+varepsilon }.$ [ABSTRACT FROM AUTHOR]

Let $L(s,pi )$ be the automorphic $L$-function associated to an automorphic irreducible cuspidal representation $pi $ of $text {GL}_m$ over $mathbb {Q}$, and let $a_{pi }(n)$ be the $n$th coefficient in its Dirichlet series expansion. In this paper we prove that if at every finite place $p$, $pi _p$ is unramified, then for any $varepsilon >0$, begin {eqnarray*} A_{pi }(x)=sum _{n leq x}a_{pi }(n) ll _{varepsilon ,pi } left {begin {array}{ccc} x^{frac {71}{192}+varepsilon }& text { if }& m=2, x^{frac {m^2-m}{m^2+1}+varepsilon }& text { if }& m geq 3. endarray }}} right . endeqnarray* }}} [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of finding the extremal element for a linear functional over a uniformly convex Banach space. We show that a unique extremal element exists and depends continuously on the linear functional, and vice versa. Using this, we simplify and clarify Ryabykh's proof that for any linear functional on a uniformly convex Bergman space with kernel in a certain Hardy space, the extremal function belongs to the corresponding Hardy space. [ABSTRACT FROM AUTHOR]

In this paper we study the regularity theory for the Poisson equation in $mathbb {R}^n$ under proper conditions. Furthermore, it will be verified that these conditions are optimal. [ABSTRACT FROM AUTHOR]

In this paper we prove a weighted version of the Khintchine inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we prove that separable subgroups have bounded packing in ambient groups. The notion bounded packing was introduced by Hruska and Wise, and, in particular, our result confirms a conjecture of theirs which states that each subgroup of a virtually polycyclic group has the bounded packing property. [ABSTRACT FROM AUTHOR]

We prove that there exists a countable family of continuous real functions whose graphs, together with their inverses, cover an uncountable square, i.e. a set of the form X × X, where X ⊆ … is uncountable. This extends Sierpiński's theorem from 1919, saying that S × S can be covered by countably many graphs of functions and inverses of functions if and only if |S| ≤ …1. Using forcing and absoluteness arguments, we also prove the existence of countably many 1-Lipschitz functions on the Cantor set endowed with the standard non-archimedean metric that cover an uncountable square. [ABSTRACT FROM AUTHOR]

The sharp trace inequality of José Escobar is extended to traces for the fractional Laplacian on …n, and a complete characterization of cases of equality is discussed. The proof proceeds via Fourier transform and uses Lieb's sharp form of the Hardy-Littlewood-Sobolev inequality. [ABSTRACT FROM AUTHOR]

Let A be a (complex, unital) semisimple Banach algebra and denote by ΩA its open spectral unit ball, that is, the set ΩA = {a ∈, A : σ(a) ⊆ D}, where σ (a) denotes the spectrum of a in A and D is the open unit disc in the complex plane. We prove that if F : ΩA → ΩA is a holomorphic map satisfying F (0) = 0 and F′ (0) = I (the identity of A), then for a in ΩA the intersection of all closed discs lying inside D and containing σ (a) equals the intersection of all closed discs lying inside D and containing σ (F (a)). When all the elements of A have an at most countable spectrum and F is biholomorphic, this implies that F preserves the convex hull of the spectrum. As an application of the same equality, we prove that if B is a semisimple Banach algebra and T : A → B is a unital surjective spectral isometry, then σ (T (a)) = σ (a) in the case when σ (a) has exactly two elements. [ABSTRACT FROM AUTHOR]

We give a local deformation theoretic proof of Farkas' conjecture, first proved by Grushevsky and Salvati Manni, that a complex principally polarized abelian variety (ppav) of dimension 4 whose theta divisor has an isolated double point of rank 3 at a point of order two is a Jacobian of a smooth curve of genus 4. The basis of this proof is Beauville's result that a 4 dimensional ppav is a non-hyperelliptic Jacobian if and only if some symmetric translate of the theta divisor has singular locus which either consists of precisely two distinct conjugate singularities {±x} or has an isolated singular point which is a limit of two distinct conjugate singularities. We establish an explicit local normal form for the theta function near an isolated double point of rank 3 at a point of order two, which implies the point is such a limit (after translation to the origin), i.e. has a small deformation within the family defined by the universal theta function whose nearby singularities include two conjugate ordinary double points (odp's). The existence of such a deformation depends only on the facts that the theta function is even, a general theta divisor is smooth, and a general singular theta divisor has only odp's, also proved by Beauville in dimension 4. The argument yields a similar result, also proved by Grushevsky and Salvati Manni, for ppav's of dimension g > 4 whose theta divisor has an isolated double point of rank (g-1), i.e. corank one, at a point of order two. [ABSTRACT FROM AUTHOR]

We prove, subject to certain hypotheses, that a positive proportion of the a-points of the Riemann zeta-function and Dirichlet L-functions with primitive characters are simple and discuss corresponding results for other functions in the Selberg class. We also prove an unconditional result of this type for the a-points in fixed strips to the right of the line ℜs = 1/2. [ABSTRACT FROM AUTHOR]