Arithmetic properties of 7-regular partitions

Let \(b_{\ell }(n)\) denote the number of \(\ell \)-regular partitions of n. By employing the modular equation of seventh order, we establish the following congruence for \(b_{7}(n)\) modulo powers of 7: for \(n\ge 0\) and \(j\ge 1\),

$$\begin{aligned} b_{7}\left( 7^{2j-1}n+\frac{3\cdot 7^{2j}-1}{4}\right) \equiv 0 \pmod {7^j}. \end{aligned}$$

We also find some infinite families of congruences modulo 2 and 7 satisfied by \(b_{7}(n)\).

     

On $$(\ell,m)$$-regular partitions with distinct parts

Let \(a_{\ell ,m}(n)\) denote the number of \((\ell ,m)\)-regular partitions of a positive integer n into distinct parts, where \(\ell \) and m are relatively primes. In this paper, we establish several infinite families of congruences modulo 2 for \(a_{3,5}(n)\). For example,

$$\begin{aligned} a_{3, 5}\left(2^{6\alpha +4}5^{2\beta }n+\frac{ 2^{6\alpha +3}5^{2\beta +1}-1}{3}\right) \equiv 0 , \end{aligned}$$

where \(\alpha , \beta \ge 0\).

     

Groups with a ternary equivalence relation

We consider in a group \((G,\cdot )\) the ternary relation

$$\begin{aligned} \kappa := \{(\alpha , \beta , \gamma ) \in G^3 \ | \ \alpha \cdot \beta ^{-1} \cdot \gamma = \gamma \cdot \beta ^{-1} \cdot \alpha \} \end{aligned}$$

and show that \(\kappa \) is a ternary equivalence relation if and only if the set \( \mathfrak Z \) of centralizers of the group G forms a fibration of G (cf. Theorems 2, 3). Therefore G can be provided with an incidence structure

$$\begin{aligned} \mathfrak G:= \{\gamma \cdot Z \ | \ \gamma \in G , Z \in \mathfrak Z(G) \}. \end{aligned}$$

We study the automorphism group of \((G,\kappa )\), i.e. all permutations \(\varphi \) of the set G such that \( (\alpha , \beta , \gamma ) \in \kappa \) implies \((\varphi (\alpha ),\varphi (\beta ),\varphi (\gamma ))\in \kappa \). We show \(\mathrm{Aut}(G,\kappa )=\mathrm{Aut}(G,\mathfrak G)\), \(\mathrm{Aut} (G,\cdot ) \subseteq \mathrm{Aut}(G,\kappa )\) and if \( \varphi \in \mathrm{Aut}(G,\kappa )\) with \(\varphi (1)=1\) and \(\varphi (\xi ^{-1})= (\varphi (\xi ))^{-1}\) for all \(\xi \in G\) then \(\varphi \) is an automorphism of \((G,\cdot )\). This allows us to prove a representation theorem of \(\mathrm{Aut}(G,\kappa )\) (cf. Theorem 6) and that for \(\alpha \in G \) the maps

$$\begin{aligned} \tilde{\alpha }\ : \ G \rightarrow G;~ \xi \mapsto \alpha \cdot \xi ^{-1} \cdot \alpha \end{aligned}$$

of the corresponding reflection structure \((G, \widetilde{G})\) (with \( \tilde{G} := \{\tilde{\gamma }\ | \ \gamma \in G \}\)) are point reflections. If \((G ,\cdot )\) is uniquely 2-divisible and if for \(\alpha \in G\), \(\alpha ^{1\over 2}\) denotes the unique solution of \(\xi ^2=\alpha \) then with \(\alpha \odot \beta := \alpha ^{1\over 2} \cdot \beta \cdot \alpha ^{1\over 2}\), the pair \((G,\odot )\) is a K-loop (cf. Theorem 5).

     

Congruences for (3,11)-regular bipartitions modulo 11

In this note we investigate the function \(B_{k,\ell }(n)\), which counts the number of \((k,\ell )\)-regular bipartitions of n. We shall prove an infinite family of congruences modulo 11: for \(\alpha \ge 2\) and \(n\ge 0\),

$$\begin{aligned} B_{3,11}\left( 3^{\alpha }n+\frac{5\cdot 3^{\alpha -1}-1}{2}\right) \equiv 0\ (\mathrm{mod\ }11). \end{aligned}$$

     

Congruences modulo 16, 32, and 64 for Andrews’s singular overpartitions

In a recent work, Andrews gave a definition of combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters k and i, can be enumerated by the function \(\overline{C}_{k,i}(n) \) which denotes the number of overpartitions of n in which no part is divisible by k and only parts \(\equiv \pm i \ (\mathrm{mod}\ k)\) may be overlined. Andrews, Chen, Hirschhorn and Sellers, and Ahmed and Baruah discovered numerous congruences modulo 2, 3, 4, 8, and 9 for \(\overline{C}_{3,1}(n)\). In this paper, we prove a number of congruences modulo 16, 32, and 64 for \(\overline{C}_{3,1}(n)\).     

Periodic analogues of Dedekind sums and transformation formulas of Eisenstein series

Let \(\bar{p}(n)\) denote the number of overpartitions of n. Fortin et al. and Hirschhorn and Sellers established some congruences modulo powers of 2 for \(\bar{p}(n)\). Recently, Xia and Yao found several congruences modulo powers of 2 and 3. In particular, they proved that \(\bar{p}(96n+12)\equiv 0 \ (\mathrm{mod}\ 9)\) and \(\bar{p}(24n+19)\equiv 0\ (\mathrm{mod\ }27)\). In this paper, we generalize the two congruences and establish several new infinite families of congruences modulo 9 and 27 for \(\bar{p}(n)\). Furthermore, we prove some strange congruences modulo 9 and 27 for \(\bar{p}(n)\) by employing some results due to Cooper et al. For example, we prove that for \(k\ge 0\), \(\bar{p}(4^{k+1})\equiv 2^{k+3}+6(-1)^k\ (\mathrm{mod} \ 27) \) and \(\bar{p}\left( 7^{2k}\right) \equiv 2-2k\ (\mathrm{mod}\ 9)\). We also present two conjectures on congruences for \(\bar{p}(n)\).     

Congruences for 7 and 49-regular partitions modulo powers of 7

Let \(b_{k}(n)\) denote the number of k-regular partitions of n. In this paper, we prove Ramanujan-type congruences modulo powers of 7 for \(b_{7}(n)\) and \(b_{49}(n)\). For example, for all \(j\ge 1\) and \(n\ge 0\), we prove that

$$\begin{aligned} b_{7}\Bigg (7^{2j-1}n+\frac{3\cdot 7^{2j-1}-1}{4}\Bigg )\equiv 0\pmod {7^{j}} \end{aligned}$$

and

$$\begin{aligned} b_{49}\Big (7^{j}n+7^{j}-2\Big )\equiv 0\pmod {7^{j}}. \end{aligned}$$

     

Congruences for $$\ell $$-regular overpartitions and Andrews’ singular overpartitions

Let \(\overline{A}_{\ell }(n)\) be the number of overpartitions of n into parts not divisible by \(\ell \). In a recent paper, Shen calls the overpartitions enumerated by the function \(\overline{A}_{\ell }(n)\) as \(\ell \)-regular overpartitions. In this paper, we find certain congruences for \(\overline{A}_{\ell }(n)\), when \(\ell =4, 8\), and 9. Recently, Andrews introduced the partition function \(\overline{C}_{k, i}(n)\), called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts \(\equiv \pm i\pmod {k}\) may be over-lined. He also proved that \(\overline{C}_{3, 1}(9n+3)\) and \(\overline{C}_{3, 1}(9n+6)\) are divisible by 3. In this paper, we prove that \(\overline{C}_{3, 1}(12n+11)\) is divisible by 144 which was conjectured to be true by Naika and Gireesh.     

On $$\ell $$-regular bipartitions modulo $$\ell $$

Let \(B_\ell (n)\) denote the number of \(\ell \)-regular bipartitions of n. In this paper, we prove several infinite families of congruences satisfied by \(B_\ell (n)\) for \(\ell \in {\{5,7,13\}}\). For example, we show that for all \(\alpha >0\) and \(n\ge 0\),

$$\begin{aligned} B_5\left( 4^\alpha n+\frac{5\times 4^\alpha -2}{6}\right)\equiv & {} 0 \ (\text {mod}\ 5),\\ B_7\left( 5^{8\alpha }n+\displaystyle \frac{5^{8\alpha }-1}{2}\right)\equiv & {} 3^\alpha B_7(n)\ (\text {mod}\ 7) \end{aligned}$$

and

$$\begin{aligned} B_{13}\left( 5^{12\alpha }n+5^{12\alpha }-1\right) \equiv B_{13}(n)\ (\text {mod}\ 13). \end{aligned}$$

     

Congruences modulo powers of 5 for two restricted bipartitions

Let \(B_{5}(n)\) denote the number of 5-regular bipartitions of n. We establish some Ramanujan-type congruences like \(B_{5}(4n+3) \equiv 0\) (mod 5) and many infinite families of congruences for \(B_{5}(n)\) modulo higher powers of 5 such as

$$\begin{aligned} B_{5}\left( 5^{2k-1}n+\frac{2\cdot 5^{2k-1}-1}{3}\right) \equiv 0 \pmod {5^k}. \end{aligned}$$

We also apply the same method to obtain some similar results for another type of bipartition function. Meanwhile, we give a new interesting interlinked q-series identity related with Rogers–Ramanujan continued fraction, which answers a question of M. Hirschhorn.

     

Arithmetic properties of overpartition triples

Let p_3(n) be the number of overpartition triples of n. By elementary series manipulations,we establish some congruences for p_3(n) modulo small powers of 2, such as p_3(16 n + 14) ≡ 0(mod 32), p_3(8 n + 7) ≡ 0(mod 64).We also find many arithmetic properties for p_3(n) modulo 7, 9 and 11, involving the following infinite families of Ramanujan-type congruences: for any integers α≥ 1 and n ≥ 0, we have p_3 (3~(2α+1)(3n + 2))≡ 0(mod 9 · 2~4), p_3(4~(α-1)(56 n + 49)) ≡ 0(mod 7),p_3 (7~(2α+1)(7 n + 3))≡ p_3 (7~(2α+1)(7 n + 5))≡ p_3 (7~(2α+1)(7 n + 6))≡ 0(mod 7),and for r ∈ {1, 2, 3, 4, 5, 6},p_3(11 · 7~(4α-1)(7 n + r)≡ 0(mod 11).     

Congruences for 5-regular partitions modulo powers of 5

Let \(b_{5}(n)\) denote the number of 5-regular partitions of n. We find the generating functions of \(b_{5}(An+B)\) for some special pairs of integers (A, B). Moreover, we obtain infinite families of congruences for \(b_{5}(n)\) modulo powers of 5. For example, for any integers \(k\ge 1\) and \(n\ge 0\), we prove that

$$\begin{aligned} b_{5}\left( 5^{2k-1}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}) \end{aligned}$$

and

$$\begin{aligned} b_{5}\left( 5^{2k}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}). \end{aligned}$$

     

Congruences for Andrews’ (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">i</Emphasis>)-singular overpartitions

Andrews recently defined new combinatorial objects which he called (k, i)-singular overpartitions and proved that they are enumerated by \(\overline{C}_{k,i}(n)\) which is the number of overpartitions of n in which no part is divisible by k and only the parts \(\equiv \pm i \pmod {k}\) may be overlined. Andrews further showed that \(\overline{C}_{3,1}(n)\) satisfies some Ramanujan-type congruences modulo 3. In this paper, we show that for any pair (k, i), \(\overline{C}_{k,i}(n)\) satisfies infinitely many Ramanujan-type congruences modulo any power of prime coprime to 6k. We also show that for an infinite family of k, the value \(\overline{C}_{3k,k}(n)\) is almost always even. Finally, we investigate the parity of \(\overline{C}_{4k,k}\).     

Ramanujan-type congruences for overpartitions modulo 16

Let \(\bar{p}(n)\) denote the number of overpartitions of \(n\). Recently, Fortin–Jacob–Mathieu and Hirschhorn–Sellers independently obtained 2-, 3- and 4-dissections of the generating function for \(\bar{p}(n)\) and derived a number of congruences for \(\bar{p}(n)\) modulo 4, 8 and 64 including \(\bar{p}(8n+7)\equiv 0 \pmod {64}\) for \(n\ge 0\). In this paper, we give a 16-dissection of the generating function for \(\bar{p}(n)\) modulo 16 and show that \(\bar{p}(16n+14)\equiv 0\pmod {16}\) for \(n\ge 0\). Moreover, using the \(2\)-adic expansion of the generating function for \(\bar{p}(n)\) according to Mahlburg, we obtain that \(\bar{p}(\ell ^2n+r\ell )\equiv 0\pmod {16}\), where \(n\ge 0\), \(\ell \equiv -1\pmod {8}\) is an odd prime and \(r\) is a positive integer with \(\ell \not \mid r\). In particular, for \(\ell =7\) and \(n\ge 0\), we get \(\bar{p}(49n+7)\equiv 0\pmod {16}\) and \(\bar{p}(49n+14)\equiv 0\pmod {16}\). We also find four congruence relations: \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n) \pmod {16}\) for \(n\ge 0\), \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {32}\) where \(n\) is not a square of an odd positive integer, \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {64}\) for \(n\not \equiv 1,2,5\pmod {8}\) and \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {128}\) for \(n\equiv 0\pmod {4}\).     

On Lipschitz continuity of solutions of hyperbolic Poisson’s equation

In this paper, we investigate solutions of the hyperbolic Poisson equation \(\Delta _{h}u(x)=\psi (x)\), where \(\psi \in L^{\infty }(\mathbb {B}^{n}, {\mathbb R}^n)\) and

$$\begin{aligned} \Delta _{h}u(x)= (1-|x|^2)^2\Delta u(x)+2(n-2)\left( 1-|x|^2\right) \sum _{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \end{aligned}$$

is the hyperbolic Laplace operator in the n-dimensional space \(\mathbb {R}^n\) for \(n\ge 2\). We show that if \(n\ge 3\) and \(u\in C^{2}(\mathbb {B}^{n},{\mathbb R}^n) \cap C(\overline{\mathbb {B}^{n}},{\mathbb R}^n )\) is a solution to the hyperbolic Poisson equation, then it has the representation \(u=P_{h}[\phi ]-G_{ h}[\psi ]\) provided that \(u\mid _{\mathbb {S}^{n-1}}=\phi \) and \(\int _{\mathbb {B}^{n}}(1-|x|^{2})^{n-1} |\psi (x)|\,d\tau (x)<\infty \). Here \(P_{h}\) and \(G_{h}\) denote Poisson and Green integrals with respect to \(\Delta _{h}\), respectively. Furthermore, we prove that functions of the form \(u=P_{h}[\phi ]-G_{h}[\psi ]\) are Lipschitz continuous.

     

Book reviews

We consider the following singularly perturbed nonlocal elliptic problem

$$\begin{aligned} -\left( \varepsilon ^{2}a+\varepsilon b\displaystyle \int _{\mathbb {R}^{3}}|\nabla u|^{2}dx\right) \Delta u+V(x)u=\displaystyle \varepsilon ^{\alpha -3}(W_{\alpha }(x)*|u|^{p})|u|^{p-2}u, \quad x\in \mathbb {R}^{3}, \end{aligned}$$

where \(\varepsilon >0\) is a parameter, \(a>0,b\ge 0\) are constants, \(\alpha \in (0,3)\), \(p\in [2, 6-\alpha )\), \(W_{\alpha }(x)\) is a convolution kernel and V(x) is an external potential satisfying some conditions. By using variational methods, we establish the existence and concentration of positive ground state solutions for the above equation.

     

A characterization of nonlinear Hölder seminorm preserving bijections

For a compact metric space (X, d) and \(\alpha \in (0,1)\), let \(\mathrm{Lip}^\alpha (X)\) be the linear space of all complex-valued functions f on X satisfying

Open image in new window

and \(\mathrm{lip}^\alpha (X)\) be the subspace of \(\mathrm{Lip}^\alpha (X)\) consisting of functions f with \(\lim \frac{f(x)-f(y)}{d^\alpha (x,y)} =0\) as \(d(x,y) \rightarrow 0\). In this paper, we give a characterization of a bijective map \(T:\mathrm{lip}^\alpha (X)\longrightarrow \mathrm{lip}^\alpha (Y)\), not necessarily linear, which is an isometry with respect to the Hölder seminorm \(L(\cdot )\). It is shown that there exist \(K_0>0\), a surjective map \(\Psi : Y \longrightarrow X\) with \(d^\alpha (y,z)= K_0 \, d^\alpha (\Psi (y),\Psi (z))\) for all \(y,z\in Y\), and a function \(\Lambda : \mathrm{lip}^\alpha (X) \longrightarrow {\mathbb {C}}\) (which is linear or real-linear if T is so) such that either

$$\begin{aligned} Tf(y)= T0(y)+\overline{\tau } K_0\, f(\Psi (y))+\Lambda (f)\quad (f\in \mathrm{lip}^\alpha (X), y\in Y) \end{aligned}$$

or

$$\begin{aligned} Tf(y)= T0(y)+\overline{\tau } K_0 \,\overline{f(\Psi (y))}+ \Lambda (f)\quad (f\in \mathrm{lip}^\alpha (X), y\in Y), \end{aligned}$$

where \(\tau =e^{i\theta }\) for some \(\theta \in [0,\pi )\).

     

On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where

$$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$

is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and

$$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$

then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

     

Multiplicity and bifurcation of positive solutions for nonhomogeneous semilinear fractional Laplacian problems

In this paper we consider the following nonhomogeneous semilinear fractional Laplacian problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u+u=\lambda (f(x,u)+h(x)) \,\, \text {in}\,\, \mathbb {R}^N,\\ u\in H^s(\mathbb {R}^N), u>0\,\, \text {in}\,\, \mathbb {R}^N, \end{array}\right. } \end{aligned}$$

where \(\lambda >0\) and \(\lim _{|x|\rightarrow \infty }f(x,u)=\overline{f}(u)\) uniformly on any compact subset of \([0,\infty )\). We prove that under suitable conditions on f and h, there exists \(0<\lambda ^*<+\infty \) such that the problem has at least two positive solutions if \(\lambda \in (0,\lambda ^*)\), a unique positive solution if \(\lambda =\lambda ^*\), and no solution if \(\lambda >\lambda ^*\). We also obtain the bifurcation of positive solutions for the problem at \((\lambda ^*,u^*)\) and further analyse the set of positive solutions.

     

Zeta functions and asymptotic additive bases with some unusual sets of primes

Fix \(\delta \in (0,1]\), \(\sigma _0\in [0,1)\) and a real-valued function \(\varepsilon (x)\) for which \(\varlimsup _{x\rightarrow \infty }\varepsilon (x)\leqslant 0\). For every set of primes \(\mathcal {P}\) whose counting function \(\pi _\mathcal {P}(x)\) satisfies an estimate of the form

$$\begin{aligned} \pi _\mathcal {P}(x)=\delta \,\pi (x)+O\bigl (x^{\sigma _0+\varepsilon (x)}\bigr ), \end{aligned}$$

we define a zeta function \(\zeta _\mathcal {P}(s)\) that is closely related to the Riemann zeta function \(\zeta (s)\). For \(\sigma _0\leqslant \frac{1}{2}\), we show that the Riemann hypothesis is equivalent to the non-vanishing of \(\zeta _\mathcal {P}(s)\) in the region \(\{\sigma >\frac{1}{2}\}\).

For every set of primes \(\mathcal {P}\) that contains the prime 2 and whose counting function satisfies an estimate of the form

$$\begin{aligned} \pi _\mathcal {P}(x)=\delta \,\pi (x)+O\bigl ((\log \log x)^{\varepsilon (x)}\bigr ), \end{aligned}$$

we show that \(\mathcal {P}\) is an exact asymptotic additive basis for \(\mathbb {N}\), i.e. for some integer \(h=h(\mathcal {P})>0\) the sumset \(h\mathcal {P}\) contains all but finitely many natural numbers. For example, an exact asymptotic additive basis for \(\mathbb {N}\) is provided by the set

$$\begin{aligned} \{2,547,1229,1993,2749,3581,4421,5281\ldots \}, \end{aligned}$$

which consists of 2 and every hundredth prime thereafter.