Let $p_1,p_2,\cdots$ be the sequence of all primes in ascending order. Using explicit estimates from the prime number theory, we show that if $k\ge 5$, then $$(p_{k+1}-1)!\mid \left(\frac{1}{2}(p_{k+1}-1)\right)!p_k!,$$ which improves a previous result of the second author.

In a stunning new advance towards the Prime k-Tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple $\left(x\right)={gx+h_j}_{j=1}^k$ of linear forms in ℤ[x], the set $\left(n\right)={gn+h_j}_{j=1}^k$ contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that $\left(n\right)={gn+h_j}_{j=1}^k$ contains at least m consecutive primes for infinitely many n ∈ ℕ. We answer an old question of Erdős and Turán by producing strings of m + 1 consecutive primes whose successive gaps ${\delta }_1,...,{\delta }_m$ form an increasing...

Let $K/\mathbb{Q}$ be an algebraic number field of class number one and let ${𝒪}_K$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in ${𝒪}_K$ under the assumption of the $abc$-conjecture for number fields.

The goal of the article is to develop a theory dual to that of support in the derived category $\mathrm{D}\left(R\right)$. This is done by introducing ‘big’ and ‘small’ cosupport for complexes that are different from the cosupport in D. J. Benson, S. B. Iyengar, H. Krause (2012). We give some properties for cosupport that are similar, or rather dual, to those of support for complexes, study some relations between ‘big’ and ‘small’ cosupport and give some comparisons of support and cosupport. Finally, we investigate...

For a double complex $(A,d^{\text{'}},d^{\text{'}\text{'}})$, we show that if it satisfies the $d^{\text{'}}{d}^{\text{'}\text{'}}$-lemma and the spectral sequence $\left\{E_r^{p,q}\right\}$ induced by $A$ does not degenerate at $E_0$, then it degenerates at $E_1$. We apply this result to prove the degeneration at $E_1$ of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of $d^{\text{'}}{d}^{\text{'}\text{'}}$-lemma.

We consider certain class of second order nonlinear nonautonomous delay differential equations of the form $$a\left(t\right)x^{\text{'}\text{'}}+b\left(t\right)g(x,x^{\text{'}})+c\left(t\right)h\left(x(t-r)\right)m\left(x^{\text{'}}\right)=p(t,x,x^{\text{'}})$$ and $${\left(a\left(t\right)x^{\text{'}}\right)}^{\text{'}}+b\left(t\right)g(x,x^{\text{'}})+c\left(t\right)h\left(x(t-r)\right)m\left(x^{\text{'}}\right)=p(t,x,x^{\text{'}}),$$ where $a$, $b$, $c$, $g$, $h$, $m$ and $p$ are real valued functions which depend at most on the arguments displayed explicitly and $r$ is a positive constant. Different forms of the integral inequality method were used to investigate the boundedness of all solutions and their derivatives. Here, we do not require construction of the Lyapunov-Krasovski functional to establish our results....

We consider elementary operators $x\mapsto {\sum }_{j=1}^n{a}_{j}x{b}_j$, acting on a unital Banach algebra, where $a_j$ and $b_j$ are separately commuting families of generalized scalar elements. We give an ascent estimate and a lower bound estimate for such an operator. Additionally, we give a weak variant of the Fuglede-Putnam theorem for an elementary operator with strongly commuting families $a_j$ and $b_j$, i.e. $a_j=a_j^{\text{'}}+i{a}_j^{\text{'}\text{'}}$ ($b_j=b_j^{\text{'}}+i{b}_j^{\text{'}\text{'}}$), where all $a_j^{\text{'}}$ and $a_j^{\text{'}\text{'}}$ ($b_j^{\text{'}}$ and $b_j^{\text{'}\text{'}}$) commute. The main tool is an L¹ estimate of the Fourier transform of a certain class...

In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for $S^1$-valued maps $m^{\text{'}}$ (the magnetization) of two variables $x^{\text{'}}$: $E_{\epsilon }\left(m^{\text{'}}\right)=\epsilon \int {|{\nabla }^{\text{'}}·m^{\text{'}}|}^{2}d{x}^{\text{'}}+\frac{1}{2}\int {\left||{\nabla }^{\text{'}}{|}^{-1/2}{\nabla }^{\text{'}}·m^{\text{'}}\right|}^{2}d{x}^{\text{'}}$. We are interested in the behavior of minimizers as $\epsilon \to 0$. They are expected to be $S^1$-valued maps $m^{\text{'}}$ of vanishing distributional divergence ${\nabla }^{\text{'}}·m^{\text{'}}=0$, so that appropriate boundary conditions enforce line discontinuities. For finite $\epsilon >0$, these line discontinuities are approximated by smooth transition layers, the so-called Néel...

We determine the duals of the homogeneous matrix-weighted Besov spaces ${Ḃ}_p^{\alpha q}\left(W\right)$ and ${ḃ}_p^{\alpha q}\left(W\right)$ which were previously defined in [5]. If W is a matrix $A_p$ weight, then the dual of ${Ḃ}_p^{\alpha q}\left(W\right)$ can be identified with ${Ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(W^{-p^{\text{'}}/p}\right)$ and, similarly, $\left[{ḃ}_p^{\alpha q}\left(W\right)\right]*\approx {ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(W^{-p^{\text{'}}/p}\right)$. Moreover, for certain W which may not be in the $A_p$ class, the duals of ${Ḃ}_p^{\alpha q}\left(W\right)$ and ${ḃ}_p^{\alpha q}\left(W\right)$ are determined and expressed in terms of the Besov spaces ${Ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(A_Q^{-1}\right)$ and ${ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(A_Q^{-1}\right)$, which we define in terms of reducing operators ${A_Q}_Q$ associated with W. We also develop the basic theory of these reducing operator Besov spaces....

In this note, we show that the counting function of the number of composite positive integers $n\le x$ such that $\beta \left(n\right)={\sum }_{p\mid n}p$ is a prime is of order of magnitude at least $x/{(logx)}^3$ and at most $x/logx$.

Let ⊆ be ideals of a Noetherian ring R, and let N be a non-zero finitely generated R-module. The set Q̅*(,N) of quintasymptotic primes of with respect to N was originally introduced by McAdam. Also, it has been shown by Naghipour and Schenzel that the set $A{*}_a(,N):={\bigcup }_{n\ge 1}As{s}_{R}R/{\left(ⁿ\right)}_a^{\left(N\right)}$ of associated primes is finite. The purpose of this paper is to show that the topology on N defined by ${{\left(ⁿ\right)}_a^{\left(N\right)}{:}_R⟨⟩}_{n\ge 1}$ is finer than the topology defined by ${{\left(ⁿ\right)}_a^{\left(N\right)}}_{n\ge 1}$ if and only if $A{*}_a(,N)$ is disjoint from the quintasymptotic primes of with respect to N. Moreover,...

We study the existence of solutions of the system $$\left\{\begin{array}{c}{\left({\phi }_1\left(u_1^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}=f_1(t,u_1\left(t\right),u_2\left(t\right),u_1^{\text{'}}\left(t\right),u_2^{\text{'}}\left(t\right)),\text{a.e.}t\in [0,T],{\left({\phi }_2\left(u_2^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}=f_2(t,u_1\left(t\right),u_2\left(t\right),u_1^{\text{'}}\left(t\right),u_2^{\text{'}}\left(t\right)),\text{a.e.}t\in [0,T],\hfill \end{array}\right.$$ submitted to nonlinear coupled boundary conditions on $[0,T]$ where ${\phi }_1,{\phi }_2:(-a,a)\to ℝ$, with $0<a<+\infty$, are two increasing homeomorphisms such that ${\phi }_1\left(0\right)={\phi }_2\left(0\right)=0$, and $f_i:[0,T]\times {ℝ}^4\to ℝ$, $i\in \{1,2\}$ are two $L^1$-Carathéodory functions. Using some new conditions and Schauder fixed point Theorem, we obtain solvability result.

Viene dimostrata l'esistenza di soluzioni del problema di Darboux per l'equazione iperbolica $z_{xy}^{\mathrm{\prime \prime }}=f(x,y,z,Z_x^{\prime },z_y)$ sul planiquarto $x\ge 0$, $y\ge 0$. Qui, $f$ è una funzione continua, con valori in uno spazio Banach che soddisfano alcune condizioni di regolarità espresse in termini della misura di non-compattezza $\alpha$.