This note uses a diophantine problem arising in elementary geometry as a prerequisite to illustrate some theory of elliptic curves. As a typical example, Proposition~\ref {5.3} and Theorem~\ref {main} determine the exact set of ration\-al numbers for which the specialization homomorphism from the torsion free rank~2 group of rational points on some elliptic curve over $\mathbb {Q}(t)$, is well defined and injective.

In this paper, we build some variations of multiple zeta values and investigate their relations. Among other things, we prove that \[ \sum _{\substack {|\boldsymbol {\alpha }|=m+r\\ 1\leq k_1\lt k_2\lt \cdots \lt k_r}} \!\!\!\!\!\!\!\!k_1^{-\alpha _1}k_2^{-\alpha _2} \cdots k_r^{-\alpha _r}(k_r-p)^{-1} \] can be evaluated as a linear combination of $\zeta (r), \zeta (r-1), \ldots , \zeta (r-p+1)$ for $r\geq p+1$. In particular, for $r\geq 2$, \[ \sum _{\substack {|{\bf \alpha }|=m+r\\ 1\leq k_1\lt k_2\lt \cdots \lt k_r}} \!\!\!\!\!\!\!\!k_1^{-\alpha _1}k_2^{-\alpha _2} \cdots k_r^{-\alpha _r}(k_r-1)^{-1}=\zeta (r), \] which may be compared to the well-known sum formula. A similar discussion leads to the twisted sum formula.

Let $C$ and $D$ be a pair of crumpled $n$-cubes and $h$ a homeo\-morphism of Bd$C$ to Bd$D$ for which there exists a map $f_h: C\to D$ such that $f_h\mid$ Bd$C=h$ and $f_{h}^{-1}$Bd$D$=Bd$C $. In our view, the presence of such a triple $(C,D,h)$ suggests that $C$ is ``at least as wild as" $D$. The collection $\mathscr {W}_n$ of all such triples is the subject of this paper. If $(C,D,h)\in \mathscr {W}_n$, but there is no homeomorphism such that $D$ is at least as wild as $C$, we say that $C$ is ``strictly wilder than" $D$. The latter concept imposes a partial order on the collection of crumpled $n$-cubes. Here, we study features of these wildness comparisons, and we present certain attributes of crumpled cubes that are preserved by the maps arising when $(C,D,h) \in \mathscr {W}_n$. This effort may be viewed as an initial way of classifying the wildness of crumpled cubes.

In this paper, we first prove that the iterates of a mean nonexpansive map defined on a weakly compact, convex set converge weakly to a fixed point in the presence of Opial's property and asymptotic regularity at a point. Next, we prove the analogous result for closed, convex (not necessarily bounded) subsets of uniformly convex Opial spaces. These results generalize the classical theorems for nonexpansive maps of Browder and Petryshyn in Hilbert space and Opial in reflexive spaces, satisfying Opial's condition.

We introduce the windowed Fourier transform connected with some singular partial differential operators defined on the half plane $\left [0,+\infty \right [\,\times \mathbb {R}$. Then, we investigate localization operators and show that these operators are not only bounded but also in the Shatten-von Neumann class. We give a trace formula when the symbol function is a nonnegative function.

We investigate when radicals $\mathfrak {F}$ satisfy Amit\-sur's property on skew polynomials of derivation type, namely, $\mathfrak {F}(R[x;\delta ])=(\mathfrak {F}(R[x;\delta ])\cap R)[x;\delta ].$ In particular, we give a new argument that the Brown-McCoy radical has this property. We also give a new characterization of the prime radical of $R[x;\delta ]$.

In this paper, we investigate the maximal independence polynomials of some popular graph configurations. Through careful analysis, some of the polynomials under study are shown to be Chebyshev, which helps characterize polynomial properties such as unimodality, log-concavity and real-rootedness with ease and efficiency. We partially characterize the bridge path and bridge cycle graphs of wheels and fans according to their unimodality properties and propose relevant open problems. Also, to compare with the usual independence polynomials, we provide analogous results regarding the vertebrated graph, and the firecracker graph, as studied by Wang and Zhu~\cite {WZ11}.

In this paper, a type of degenerate system defined by the sum of two homogeneous vector fields is studied. By means of a blow-up technique and a classification theorem, a monodromy criterion of an isolated singular point is presented and proven. An example is given to illustrate that our result generalizes the corresponding result in~\cite {Gas00}.

We investigate the ideal $\mathcal K_{\rm up}$, $1 \leq p \leq \infty $, of unconditionally $p$-compact operators. We obtain the isometric identities $\mathcal K_{\rm up}=\mathcal K_{\rm up}\circ \mathcal K_{\rm up}$, $\mathcal K^{\max }_{\rm up}=\mathcal L^{\rm sur}_{p^*}$, $\mathcal K^{\min }_{\rm up}=\widehat {\otimes }_{/w_{p^*}}$ and $\mathcal K_{\rm up}=\mathcal N_{\rm up}^{\rm Qdual}$ and prove that, if $X^*$ has the approximation property or $Y$ has the $\mathcal K_{\rm up}$-approximation property, then $\mathcal K_{\rm up}(X, Y)$ is isometrically equal to $\mathcal K^{\min }_{\rm up}(X, Y)$, and the dual space $\mathcal K_{\rm up}(X, Y)^*$ is isometric to $(\mathcal L_{p}^{\rm inj})^*(X^*, Y^*)$. As a consequence, for every Banach space $X$, we obtain the isometric identities $\mathcal K_{\rm up}^{\max }(\ell _{1}(\Gamma ), X) =\mathcal L_{p^*}(\ell _{1}(\Gamma ), X)$, $\mathcal K_{\rm up}^{\min }(\ell _{1}(\Gamma ), X) =\ell _{\infty }(\Gamma )\widehat {\otimes }_{w_{p^*}} X$ and $\mathcal K_{\rm up}(\ell _{1}(\Gamma ), X)^* =\mathcal D_{p^*} (\ell _{\infty }(\Gamma ), X^*)$.

This paper deals with heat equations coupled via exponential boundary flux, where the solution is made up of $n$ components. Under certain monotone assumptions, necessary and sufficient conditions are obtained for simultaneous blow-up of at least two components for each initial datum. As for two components blowing up simultaneously, it is interesting that the representations of blow-up rates are quite different with respect to the different blow-up mechanisms and positions between the two components.

We compute topological entropies for a large family of automorphisms of K3 surfaces in \(\mathbb {P}^1 \times \mathbb {P}^1 \times \mathbb {P}^1\). Similarly to a result by Xie~\cite {Xie}, we find that the entropies vary in a lower semi-continuous manner as the Picard ranks of the K3 surfaces vary.

Theorems of global existence and uniqueness of a classical solution to a nonlinear differential evolutionary system with initial conditions are proved. This system is composed of one partial hyperbolic second-order equation and an ordinary subsystem with a parameter. In the proof of the theorems we use the Picard iteration method, the monotone method of lower and upper solutions, the integral form of the differential problem, weak differential inequalities and the Arzeli-Ascola lemma.

We survey the archimedean representations and Langlands parameters corresponding to holomorphic Siegel modular forms of degree~$2$. This leads to a determination of archimedean local factors for various $L$-functions and all vector-valued weights. We determine the Hodge structures that correspond to holomorphic Siegel modular forms and clarify the relationship with four-dimensional symplectic artin representations.

This paper concerns the existence of affine-periodic solutions for nonlinear systems with certain affine-periodic symmetry. The existence result is actually proved based on the existence of upper and lower solutions and the conditions on them. Some applications are also given.