The function ${\sum }_{p\le x}1/p-loglog\left(x\right)-M$ is known to change sign infinitely often, but so far all calculated values are positive. In this paper we prove that the first sign change occurs well before exp(495.702833165).

Let ℓ be a Banach sequence space with a monotone norm $\parallel ·{\parallel }_{\ell }$, in which the canonical system $\left(e_i\right)$ is a normalized symmetric basis. We give a complete isomorphic classification of Cartesian products $E_0^{\ell }\left(a\right)\times E_{\infty }^{\ell }\left(b\right)$ where $E_0^{\ell }\left(a\right)=K^{\ell }\left(exp(-p^{-1}{a}_i)\right)$ and $E_{\infty }^{\ell }\left(b\right)=K^{\ell }\left(exp\left(p{a}_i\right)\right)$ are finite and infinite ℓ-power series spaces, respectively. This classification is the generalization of the results by Chalov et al. [Studia Math. 137 (1999)] and Djakov et al. [Michigan Math. J. 43 (1996)] by using the method of compound linear topological invariants developed by...

Let E,F be Banach spaces where F = E’ or vice versa. If F has the approximation property, then the space of nuclearly entire functions of bounded type, ${\mathscr{H}}_{Nb}\left(E\right)$, and the space of exponential type functions, Exp(F), form a dual pair. The set of convolution operators on ${\mathscr{H}}_{Nb}\left(E\right)$ (i.e. the continuous operators that commute with all translations) is formed by the transposes $\phi \left(D\right){\equiv }^t\phi$, φ ∈ Exp(F), of the multiplication operators φ :ψ ↦ φ ψ on Exp(F). A continuous operator T on ${\mathscr{H}}_{Nb}\left(E\right)$ is PDE-preserving for a set ℙ ⊆...

Sia ${ℬ}_n$ un sistema linearmente viscoelastico, omogeneo ed isotropo, caratterizzato dalla funzione di memoria $g_n(t)={\sum }_{k=1}^n{B}_k\exp (-{\beta }_{k}t)$, tipica di numerosi polimeri solidi. Si dimostra che la soluzione fondamentale $E_n$ dell’operatore integrodifferenziale che descrive i moti di ${ℬ}_n$ è, in ogni punto del suo supporto, maggiorata da quella relativa ad un opportuno solido standard ${ℬ}_1$ Di conseguenza, è possibile applicare all’analisi qualitativa dei moti di ${ℬ}_n$ alcuni risultati stabiliti in [10], quali proprietà asintotiche,...

Let ℳ be a type II₁ von Neumann algebra, τ a trace in ℳ, and L²(ℳ,τ) the GNS Hilbert space of τ. If L²(ℳ,τ)₊ is the completion of the set ${ℳ}_{sa}$ of selfadjoint elements, then each element ξ ∈ L²(ℳ,τ)₊ gives rise to a selfadjoint unbounded operator $L_{\xi }$ on L²(ℳ,τ). In this note we show that the exponential exp: L²(ℳ,τ)₊ → L²(ℳ,τ), $exp\left(\xi \right)=e^{i{L}_{\xi }}$, is continuous but not differentiable. The same holds for the Cayley transform $C\left(\xi \right)=(L_{\xi }-i){(L_{\xi }+i)}^{-1}$. We also show that the unitary group $U_{ℳ}\subset L²(ℳ,\tau )$ with the strong operator topology is not an...

Assume that $\mathrm{\Omega }$, ${\mathrm{\Omega }}^{\prime }$ are planar domains and $f:\mathrm{\Omega }\stackrel{\text{onto}}{\to }{\mathrm{\Omega }}^{\prime }$ is a homeomorphism belonging to Sobolev space $W_{\text{loc}}^{1,1}(\mathrm{\Omega };{ℝ}^2)$ with finite distortion. We prove that if the distortion function $K_f$ of $f$ satisfies the condition , then the distortion function $K_{f^{-1}}$ of $f^{-1}$ belongs to $L_{\text{loc}}^1({\mathrm{\Omega }}^{\prime })$. We show that this result is sharp in sense that the conclusion fails if ${\mathrm{dist}}_{\text{EXP}}(K_f,L^{\mathrm{\infty }})=1$. Moreover, we prove that if the distortion function $K_f$ satisfies the condition ${\mathrm{dist}}_{\text{EXP}}(K_f,L^{\mathrm{\infty }})=\lambda$ for some $\lambda >0$, then $K_{f^{-1}}$ belongs to $L_{\text{loc}}^p({\mathrm{\Omega }}^{\prime })$ for every $p\in (0,\frac{1}{2\lambda })$. As special case of this result we show that if...

We introduce the algebras satisfying the $(ℬ,n)$ condition. If $\Lambda$, $\Gamma$ are algebras satisfying the $(ℬ,n)$, $(ℰ,m)$ condition, respectively, we give a construction of $(m+n)$-almost split sequences in some subcategories ${(ℬ\otimes ℰ)}^{(i_0,j_0)}$ of $mod(\Lambda \otimes \Gamma )$ by tensor products and mapping cones. Moreover, we prove that the tensor product algebra $\Lambda \otimes \Gamma$ satisfies the $({(ℬ\otimes ℰ)}^{(i_0,j_0)},n+m)$ condition for some integers $i_0$, $j_0$; this construction unifies and extends the work of A. Pasquali (2017), (2019).

We give a classification of all linear natural operators transforming $p$-vectors (i.e., skew-symmetric tensor fields of type $(p,0)$) on $n$-dimensional manifolds $M$ to tensor fields of type $(q,0)$ on $T^{A}M$, where $T^A$ is a Weil bundle, under the condition that $p\ge 1$, $n\ge p$ and $n\ge q$. The main result of the paper states that, roughly speaking, each linear natural operator lifting $p$-vectors to tensor fields of type $(q,0)$ on $T^A$ is a sum of operators obtained by permuting the indices of the tensor products of linear natural...

We give sufficient conditions implying that the projective tensor product of two Banach spaces $X$ and $Y$ has the $p$-sequentially Right and the $p$-$L$-limited properties, $1\le p<\infty$.

We show that there exists a finitely generated group of growth $\sim f$ for all functions $f:{ℝ}_{+}\to {ℝ}_{+}$ satisfying $f\left(2R\right)\le f{\left(R\right)}^2\le f\left({\eta }_{+}R\right)$ for all $R$ large enough and ${\eta }_{+}\approx 2.4675$ the positive root of $X^3-X^2-2X-4$. Set ${\alpha }_{-}=log2/log{\eta }_{+}\approx 0.7674$; then all functions that grow uniformly faster than $exp\left(R^{{\alpha }_{-}}\right)$ are realizable as the growth of a group. We also give a family of sum-contracting branched groups of growth $\sim exp\left(R^{\alpha }\right)$ for a dense set of $\alpha \in [{\alpha }_{-},1]$.

Let 1 < p < ∞, q = p/(p-1) and for $f\in L^p(0,\infty )$ define $F\left(x\right)=(1/x){ʃ}_0^{x}f\left(t\right)dt$, x > 0. Moser’s Inequality states that there is a constant $C_p$ such that $su{p}_{a\le 1}su{p}_{f\in B_p}{ʃ}_0^{\infty }exp[a{x}^q{\left|F\left(x\right)\right|}^q-x]dx=C_p$ where $B_p$ is the unit ball of $L^p$. Moreover, the value a = 1 is sharp. We observe that $F=K_1$ f where the integral operator $K_1$ has a simple kernel K. We consider the question of for what kernels K(t,x), 0 ≤ t, x < ∞, this result can be extended, and proceed to discuss this when K is non-negative and homogeneous of degree -1. A sufficient condition on K is found for...

We investigate whether the projective tensor product of two Banach spaces $X$ and $Y$ has the reciprocal Dunford–Pettis property of order $p$, $1\le p<\infty$, when $X$ and $Y$ have the respective property.

We completely determine the ${\ell }_q$ and C(K) spaces which are isomorphic to a subspace of ${\ell }_p\otimes {̂}_{\pi }C\left(\alpha \right)$, the projective tensor product of the classical ${\ell }_p$ space, 1 ≤ p < ∞, and the space C(α) of all scalar valued continuous functions defined on the interval of ordinal numbers [1,α], α < ω₁. In order to do this, we extend a result of A. Tong concerning diagonal block matrices representing operators from ${\ell }_p$ to ℓ₁, 1 ≤ p < ∞. The first main theorem is an extension of a result of E. Oja and states...

Sia ${ℬ}_n$ un sistema linearmente viscoelastico, omogeneo ed isotropo, caratterizzato dalla funzione di memoria $g_n(t)={\sum }_{k=1}^n{B}_k\exp (-{\beta }_{k}t)$, tipica di numerosi polimeri solidi. Si dimostra che la soluzione fondamentale $E_n$ dell’operatore integrodifferenziale che descrive i moti di ${ℬ}_n$ è, in ogni punto del suo supporto, maggiorata da quella relativa ad un opportuno solido standard ${ℬ}_1$ Di conseguenza, è possibile applicare all’analisi qualitativa dei moti di ${ℬ}_n$ alcuni risultati stabiliti in [10], quali proprietà asintotiche,...

The well-known Impossibility Theorem of Arrow asserts that any generalized social welfare function (GSWF) with at least three alternatives, which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a dictatorship, is necessarily non-transitive. In 2002, Kalai asked whether one can obtain the following quantitative version of the theorem: For any $ϵ>0$, there exists $\delta =\delta \left(ϵ\right)$ such that if a GSWF on three alternatives satisfies the IIA condition and its probability of...

It is well known that, in forward inference in fuzzy logic, the generalized modus ponens is guaranteed by a functional inequality called the law of $T$-conditionality. In this paper, the $T$-conditionality for $T$-power based implications is deeply studied and the concise necessary and sufficient conditions for a power based implication $I^T$ being $T$-conditional are obtained. Moreover, the sufficient conditions under which a power based implication $I^T$ is $T^{*}$-conditional are discussed, this discussions...