The goal of this paper is to compare ranks of the divisible groups $Ext (G,A)$ and $B{\rm ext}_A^1(G,A)$ whenever~$G$ is a countable torsion-free $A$-solvable group and $A$ has a right hereditary endomorphism ring.

Let $(R,\mathfrak{m} )$ be a commutative Noetherian local ring of dimension $d\geq 1$, and let $I$ be a non-nilpotent ideal of $R$ such that the ideal transform functor $D_I(-)$ is exact. In this paper, it is shown that the finitely generated flat $R$-algebra $D_I(R)$ is a Noetherian ring of dimension $n=\dim R/\Gamma _I(R)-1$. Also, it is shown that, under Zariski topologies on the sets $Spec D_I(R)$ and $Spec R/\Gamma _I(R)$, there is a homeomorphism of topological spaces: \[ \widetilde {\eta ^*}:Spec D_I(R)\longrightarrow Spec R/\Gamma _I(R)\setminus V((I+\Gamma _I(R))/\Gamma _I(R)). \]

We extend the famous diophantine Frobenius problem to a ring of polynomials over a field~$k$. Similar to the classical problem we show that the $n=2$ case of the Frobenius problem for polynomials is easy to solve. In addition, we translate a few results from the Frobenius problem over $\mathbb{Z} $ to $k[t]$ and give an algorithm to solve the Frobenius problem for polynomials over a field $k$ of sufficiently large size.

This paper deals with construction of non-uniform multiple knot B-spline wavelet basis functions (with minimal support). These wavelets are semi-orthogonal on a bounded interval. A large family of multiple knot B-spline wavelets is presented that gives a variety of basis functions with explicit formulas and locally compact supports. Moreover, the structure of this wavelet is conceptually simple and easy to implement. Finally, some examples of multiple knot B-spline wavelets are also presented.

A definition of strict oscillation of the system of two first-order linear ordinary differential equations is given. It is shown that oscillation follows from strict oscillation of its system, but strict oscillation does not follow. Sturm-type theorems are proven. Oscillatory and strict oscillatory criteria in terms of coefficients of the system are obtained.

This paper gives sufficient conditions on the sequence $\{a_n\}_{n=1}^\infty $ of positive integers to ensure that the number $\sum _{n=1}^\infty 1/{\sqrt {a_n}}$ is irrational.

Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module. In this paper, we introduce a class of modules which is analogous to that of Goldie$^*$-lifting and principally Goldie$^*$-lifting modules. The module~$M$ is called \textit {principally} $\mathcal {G}^*$-\nobreak $\delta $-\textit {lifting} if, for any $m\in M$, there exists a direct summand $N$ of $M$ such that $mR$ is $\beta ^*_{\delta }$-equivalent to $N$. We also introduce a generalization of Goldie$^*$-supplemented modules, namely, a module $M$ is said to be \textit {principally} $\mathcal {G}^*$-$\delta $-\textit {supplemented} if, for any $m\in M$, there exists a $\delta $-supplement~$N$ in~$M$ such that $mR$ is $\beta ^*_{\delta }$-equivalent to~$N$. We prove that some results of principally $\mathcal {G}^*$-lifting modules and Goldie$^*$-lifting modules can be extended to principally $\mathcal {G}^*$-\nobreak $\delta $-lifting modules for this general setting. Several properties of these modules are given, and it is shown that the class of principally $\mathcal {G}^*$-\nobreak $\delta $-lifting modules lies between the classes of principally $\delta $-lifting modules and principally $\mathcal {G}^*$-$\delta $-supplemented modules.

Let $A$ be a $C^*$-algebra and $H$ a finite dimensional $C^*$-Hopf algebra with its dual $C^*$-Hopf algebra $H^0$. Let $(\rho , u)$ be a twisted coaction of $H^0$ on $A$. We shall define the $(\rho , u, H)$-equivariant Picard group of $A$, which is denoted by $Pic _H^{\rho , u}(A)$, and discuss the basic properties of $Pic _H^{\rho , u}(A)$. Also, we suppose that $(\rho , u)$ is the coaction of $H^0$ on the unital $C^*$-algebra $A$, that is, $u=1\otimes 1^0$. We investigate the relation between $Pic (A^s )$, the ordinary Picard group of $A^s$, and $Pic _H^{\rho ^s}(A^s )$, where $A^s$ is the stable $C^*$-alge\-bra of $A$ and $\rho ^s$ is the coaction of $H^0$ on $A^s$ induced by $\rho $. Furthermore, we shall show that $Pic _{H^0}^{\hat {\rho }}(A\rtimes _{\rho , u}H)$ is isomorphic to $Pic _H^{\rho , u}(A)$, where $\widehat {\rho }$ is the dual coaction of $H$ on the twisted crossed product $A\rtimes _{\rho , u}H$ of $A$ by the twisted coaction $(\rho , u)$ of $H^0$ on $A$.

In this paper, we establish the boundedness of rough singular integrals associated to surfaces of revolution generated by two polynomial mappings on the Triebel-Lizorkin spaces and Besov spaces.

We introduce an extension of cone splines and box splines to fractional and complex orders. These new families of multivariate splines are defined in the Fourier domain along certain $s$-directional meshes and include as special cases the $3$-directional box splines~\cite {article:condat} and hex splines~\cite {article:vandeville} previously considered by Condat and Van De Ville, et al. These cone and hex splines of fractional and complex order generalize the univariate fractional and complex B-splines defined in~\cite {article:fbu, article:ub} and, e.g., investigated in~\cite {article:fm, article:mf}. Explicit time domain representations are de\-rived for these splines on $3$-directional meshes. We present some properties of these two multivariate spline families, such as recurrence, decay and refinement. Finally, we show that a bivariate hex spline and its integer lattice translates form a Riesz basis of its linear span.

For a prime $p \equiv 5 \pmod {8}$ satisfying certain conditions, we show that there exist an infinitude of K3 surfaces parameterized by certain solutions to Pell's equation $X^2 - pY^2 = 4$ in the projective $5$-space that are counterexamples to the Hasse principle explained by the Brauer-Manin obstruction. Further, these surfaces contain no zero-cycle of odd degree over~$\mathbb{Q} $. As an illustration for the main result, we show that the prime $p = 5$ satisfies all of the required conditions in the main theorem, and hence, there exist an infinitude of K3 surfaces parameterized by the Fibonacci sequence that are counterexamples to the Hasse principle explained by the Brauer-Manin obstruction.

We prove that the sequence of gaps in the sequence of prime numbers contains infinitely many runs of three terms, with the middle term exceeding both the first and third, provided that there is at least one integer $m$ exceeding $3$, and at least one set $A$ of $2^{m-2}$ integers, with infinitely many translations of this set $n+A$ such that they contain at least $m$ primes.

In this paper, we discuss the existence of multiple solutions to a second order anti-periodic boundary value problem \[ \ddot {x}(t)+M x(t)+\nabla F(t, x(t))=0\quad\mbox{almost every } t\in [0, T],\\ x(0)=-x(T) \qquad\qquad\qquad\ \, \dot {x}(0)=-\dot {x}(T) \] by using variational methods and critical point theory. Furthermore, we obtain the existence of periodic solutions for corresponding second-order differential systems.

In this paper, we show that the solution to the semilinear wave equation with a potential of quadratic type blows up in finite time. We also give an upper bound estimate for the lifespan of the solution.