Let P and Q be nonzero integers. The sequences of generalized Fibonacci and Lucas numbers are defined by U₀ = 0, U₁ = 1 and $U_{n+1}=PUₙ-Q{U}_{n-1}$ for n ≥ 1, and V₀ = 2, V₁ = P and $V_{n+1}=PVₙ-Q{V}_{n-1}$ for n ≥ 1, respectively. In this paper, we assume that P ≥ 1, Q is odd, (P,Q) = 1, Vₘ ≠ 1, and $V_r\ne 1$. We show that there is no integer x such that $Vₙ=V_{r}Vₘx²$ when m ≥ 1 and r is an even integer. Also we completely solve the equation $Vₙ=VₘV_{r}x²$ for m ≥ 1 and r ≥ 1 when Q ≡ 7 (mod 8) and x is an even integer. Then we show that when P ≡ 3 (mod 4) and...

Let $E$ and $f$ be an Eisenstein series and a cusp form, respectively, of the same weight $k\ge 2$ and of the same level $N$, both eigenfunctions of the Hecke operators, and both normalized so that $a_1\left(f\right)=a_1\left(E\right)=1$. The main result we prove is that when $E$ and $f$ are congruent mod a prime $𝔭$ (which we take in this paper to be a prime of $\overline{\mathbb{Q}}$ lying over a rational prime $p\>2$), the algebraic parts of the special values $L(E,\chi ,j)$ and $L(f,\chi ,j)$ satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions, ...

Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.

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...

Let $G$ be a group and $m$ be an integer greater than or equal to $2$. $G$ is said to be $m$-permutable if every product of $m$ elements can be reordered at least in one way. We prove that, if $G$ has a centre of finite index $z$, then $G$ is $(1+[z/2])$-permutable. More bounds are given on the least $m$ such that $G$ is $m$-permutable.

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

We show a $p$-parity result in a $D_{2p^n}$-extension of number fields $L/K$ ($p\ge 5$) for the twist $1\oplus \eta \oplus \tau$: $W(E/K,1\oplus \eta \oplus \tau )={(-1)}^{〈1\oplus \eta \oplus \tau ,X_p(E/L)〉}$, where $E$ is an elliptic curve over $K$, $\eta$ and $\tau$ are respectively the quadratic character and an irreductible representation of degree $2$ of $\mathrm{Gal}(L/K)=D_{2p^n}$, and $X_p(E/L)$ is the $p$-Selmer group. The main novelty is that we use a congruence result between ${\epsilon }_0$-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the $p$-parity conjecture...

We show that any ${\Sigma }_s$-product of at most $𝔠$-many $L\Sigma (\le \omega )$-spaces has the $L\Sigma (\le \omega )$-property. This result generalizes some known results about $L\Sigma (\le \omega )$-spaces. On the other hand, we prove that every ${\Sigma }_s$-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every ${\Sigma }_s$-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...

We prove that if $X$ is a first countable space with property $\left(DC\left({\omega }_1\right)\right)$ and with a $G_{\delta }$-diagonal then the cardinality of $X$ is at most $𝔠$. We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$.