This paper is an introduction and a survey to the concept of perfect isometries which was first introduced by Michel Broué in 1990. Our main aim is to provide proofs of numerous results scattered in the literature. On the other hand, we make some observations which did not appear anywhere before.

The class of binary Lie algebras contains the one of Malcev algebras (and so the one of Lie algebras). We provide a classification of central extensions of complex non-Malcev binary Lie algebras of dimensions either $3$ or $4$. From here, a classification of central extensions of complex non-Lie anticommutative $\mathfrak{ℭ}\mathfrak{𝔇}$-algebras of dimensions either $3$ or $4$ is also given.

This work deals with the plane sections lying over the Pascal pyramid. Any plane section can be defined using two main diagonals with parameters $\left(q_1,r_1\right)$ and $\left(q_2,r_2\right)$. We study the main case $r_1=r_2=1$, which corresponds to plane sections crossing all parallel planes to the $x$ axis and intersecting with the integer coordinates of the pyramid. We establish the recurrence relation satisfied by the sequence of numbers counting the sum of elements lying over each plane section.

We give the fine spectrum of the quadruple band matrix operator $Q\left(r,s,t,u\right)$ over $c_0$ and $c$. The matrix $Q\left(r,s,t,u\right)$ generalizes ${\mathrm{\Delta }}^3$, $D\left(r,0,0,s\right)$, $B\left(r,s,t\right)$, ${\mathrm{\Delta }}^2$, $B\left(r,s\right)$, $\mathrm{\Delta }$, right-shift and Zweier matrices, where ${\mathrm{\Delta }}^3$, $B\left(r,s,t\right)$, ${\mathrm{\Delta }}^2$, $B\left(r,s\right)$ and $\mathrm{\Delta }$ are called third-order difference, triple band, second-order difference, double band (generalized difference) and difference matrix, respectively.

The action of ring automorphisms of $ℝ\left[x,y\right]$ on real plane curves is considered. The orbits containing degree-three polynomials are computed, with one representative per orbit being selected.

Given a locally free sheaf $\mathcal{ℱ}$ on a projective space ${\mathbb{P}}_n$, a natural question is to compute the $L^2$-cohomology of its restriction to an affine subspace ${ℂ}^n={\mathbb{P}}_n\setminus {\mathbb{P}}_{n-1}$ equipped with a flat Kähler form ${\omega }_{{ℂ}^n}^{}$. This problem arises naturally when studying the geometry of moduli of monads. We identify the known construction of a natural hermitian structure on the moduli space of the associated vector bundles with the Weil–Petersson approach to a Kähler metric based on harmonic theory for Kodaira–Spencer tensors. We show that for a holomorphic family of monads on ${\mathbb{P}}_2$ or on a modification ${\tilde{\mathbb{P}}}_2$ of ${\mathbb{P}}_2$, there is a Kodaira–Spencer map providing the base with a Kähler structure.

We introduce the $\left(\varphi ,2,\alpha \right)$-bounded variation of a function, a concept that generalizes the $\varphi$-bounded variation introduced by Medvedev. We also provide a characterization to find out the $\left(\varphi ,2,\alpha \right)$-variation of a function $f$ in the sense of Riesz.

This paper is concerned with the existence and multiplicity of solutions for the following Neumann problems with mean curvature operator in the Minkowski space:

$${\left(\frac{u^{\prime }}{\sqrt{1-{u{}^{\prime }}^2}}\right)}^{\prime }+a\left(x\right)g\left(u\right)=\mu +p\left(x\right),x\in \left(0,T\right),u^{\prime }\left(0\right)=0=u^{\prime }\left(T\right),$$

where $a,p\in L^{\infty }\left(0,T\right)$, $\mu \in ℝ$, and $g\in C^1\left(ℝ\right)$ satisfies the coercivity condition $g\left(u\right)\to +\infty$ as $\left|u\right|\to +\infty$. We show the existence results of two solutions in terms of the value of the parameter $\mu$ via the shooting method.

In this work, we discuss various kinds of $\mathcal{ℐ}$-uniform convergence for sequences of functions and introduce the concepts of ${\mathcal{ℐ}}^{\ast }$-uniform convergence, $\mathcal{ℐ}$ and ${\mathcal{ℐ}}^{\ast }$-uniformly Cauchy sequences of functions in $2$-normed spaces. Then, we show the relationships between them.

We define the grand Wiener amalgam space by using the classical Wiener amalgam space and the generalized grand Lebesgue space. Moreover we study the inclusions between these spaces and some applications. Finally we prove the Hölder inequality for the grand Wiener amalgam space. We also find the associate space and dual of this space, and we prove that the grand Wiener amalgam space is not reflexive.

In this paper, we study the critical fractional Laplacian system with Choquard type nonlinearities

$$\left\{\begin{array}{cc}{\left(-\mathrm{\Delta }\right)}^{s}u=k_1\left({\int }_{{ℝ}^N}\frac{|u\left(y\right){|}^{2_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dy\right)|u{|}^{2_{\mu }^{\ast }-2}u+\frac{\alpha \gamma }{2_{\mu }^{\ast }}\left({\int }_{{ℝ}^N}\frac{|v\left(y\right){|}^{\beta }}{|x-y{|}^{\mu }}dy\right)|u{|}^{\alpha -2}u\hfill & \text{ in }{ℝ}^N,\hfill \\ {\left(-\mathrm{\Delta }\right)}^{s}v=k_2\left({\int }_{{ℝ}^N}\frac{|v\left(y\right){|}^{2_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dy\right)|v{|}^{2_{\mu }^{\ast }-2}v+\frac{\beta \gamma }{2_{\mu }^{\ast }}\left({\int }_{{ℝ}^N}\frac{|u\left(y\right){|}^{\alpha }}{|x-y{|}^{\mu }}dy\right)|v{|}^{\beta -2}v\hfill & \text{ in }{ℝ}^N,\hfill \\ u,v\in D_s\left({ℝ}^N\right),\hfill \end{array}\right.$$

where ${\left(-\mathrm{\Delta }\right)}^s$ is the well known fractional Laplacian operator, $s\in \left(0,1\right)$, $N>2s$, $k_1,k_2>0$, $\gamma \ne 0$, $\mu \in \left(0,N\right)$, $\alpha ,\beta >1$ and $2_{\mu }^{\ast }=\left(2N-\mu \right)∕\left(N-2s\right)$ is the fractional upper critical exponent in the Hardy–Littlewood–Sobolev inequality, $D_s\left({ℝ}^N\right)$ is the Hilbert space as the completion of $C_c^{\infty }\left({ℝ}^N\right)$. We show the existence and nonexistence results for the critical nonlocal system. These extend results in the literature for the case $s=1$ and $\alpha =\beta =2_{\mu }$, where $2_{\mu }=\left(2N-\mu \right)∕\left(N-2\right)$ is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. Particularly, we establish new existence and multiplicity of positive solutions for the system when $\alpha +\beta =2\cdot 2_{\mu }^{\ast }$, whose key is providing an accurate upper bound of the least energy.

We give a complete classification of contact hypersurfaces in the complex hyperbolic two-plane Grassmannian $G_2^{\ast }\left({ℂ}^{m+2}\right)$, $m\ge 2$. In fact, they are locally congruent to a tube over a totally real totally geodesic $ℍH^n$, $m=2n$, in the complex hyperbolic two-plane Grassmannian $G_2^{\ast }\left({ℂ}^{m+2}\right)$, a horosphere whose center at the infinity is singular, or another exceptional case.

We investigate a Markov chain with a state space $1,2,\dots ,r$ stopping at appearance of patterns consisting of two consecutive states. It is observed that the expected stopping times of the chain have surprising oscillating dependencies on starting positions. Analogously, the stopping probabilities also have oscillating dependencies on terminal states. In a nonstopping Markov chain the frequencies of appearances of two consecutive states are found explicitly.

For each $\alpha >0$, the $\alpha$-Bloch space consists of all analytic functions $f$ on the unit disk satisfying $\underset{\left|z\right|<1}{sup}{\left(1-|z{|}^2\right)}^{\alpha }\left|f^{\prime }\left(z\right)\right|<+\infty$. We consider the following complex integral operators, namely the $\beta$-Cesàro operator

$$C_{\beta }\left(f\right)\left(z\right)={\int }_0^z\frac{f\left(w\right)}{w{\left(1-w\right)}^{\beta }}dw$$

and its generalization, acting from the $\alpha$-Bloch space to itself, where $f\left(0\right)=0$ and $\beta \in ℝ$. We investigate the boundedness and compactness of the $\beta$-Cesàro operators and their generalizations. Also we calculate the essential norm and spectrum of these operators.

We show that if $C_u^{\ast }\left(X\right)$ is a uniform Roe algebra associated to a bounded geometry metric space $X$, then all bounded derivations on $C_u^{\ast }\left(X\right)$ are inner.

We generalize some congruences which were proved by M. Apagodu and J.-C. Liu (2020) involving trinomial coefficients, and we obtain a supercongruence for trinomial coefficients which contains a congruence proved by Apagodu and Liu (2020).

For a positive integer $n$, let

$$F_n^{\ast }\left(X,Y\right)=\prod _{k=1}^n\left(Xsin\frac{k\pi }{n}-Ycos\frac{k\pi }{n}\right).$$

In 2000 Bean and Laugesen proved that for every $n\ge 3$ the area bounded by the curve $\left|F_n^{\ast }\left(x,y\right)\right|=1$ is equal to $4^{1-1∕n}B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right)$, where $B\left(x,y\right)$ is the beta function. We provide an elementary proof of this fact based on the polar formula for the area calculation. We also prove that

$$F_n^{\ast }\left(X,Y\right)=2^{1-n}Y\sum _{\begin{array}{c}1\le k\le n\\ k\text{ is odd}\end{array}}{\left(-1\right)}^{\frac{k-1}{2}}\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)X^{n-k}{Y}^k$$

and demonstrate that ${\ell }_n=2^{n-1-{\nu }_2\left(n\right)}$ is the smallest positive integer such that the binary form ${\ell }_n{F}_n^{\ast }\left(X,Y\right)$ has integer coefficients. Here ${\nu }_2\left(n\right)$ denotes the $2$-adic order of $n$.

Let function $f$ be analytic in the unit disk $\mathbb{𝔻}$ and be normalized so that $f\left(z\right)=z+a_2{z}^2+a_3{z}^3+\cdots$ In this paper we give sharp bounds of the modulus of its second, third and fourth coefficient, if $f$ satisfies

$$|arg\left[{\left(\frac{z}{f\left(z\right)}\right)}^{1+\alpha }{f}^{\prime }\left(z\right)\right]\parallel <\gamma \frac{1}{2}\pi \left(z\in \mathbb{𝔻}\right)$$

for $0<\alpha <1$ and $0<\gamma \le 1$.

Let $P\subset A$ be an inclusion of $\sigma$-unital $C^{\ast }$-algebras with a finite index in the sense of Pimsner–Popa. Then we introduce the Rokhlin property for a conditional expectation $E$ from $A$ onto $P$ and show that if $A$ is simple and satisfies any of the property $\left(1\right)$–$\left(12\right)$ listed in the below, and $E$ has the Rokhlin property, then so does $P$:

1. simplicity;

2. nuclearity;

3. $C^{\ast }$-algebras that absorb a given strongly self-absorbing $C^{\ast }$-algebra $\mathcal{𝒟}$;

4. $C^{\ast }$-algebras of stable rank one;

5. $C^{\ast }$-algebras of real rank zero;

6. $C^{\ast }$-algebras of nuclear dimension at most $n$, where $n\in {\mathbb{Z}}^{+}$;

7. $C^{\ast }$-algebras of decomposition rank at most $n$, where $n\in {\mathbb{Z}}^{+}$;

8. separable simple $C^{\ast }$-algebras that are stably isomorphic to AF algebras;

9. separable simple $C^{\ast }$-algebras that are stably isomorphic to AI algebras;

10. separable simple $C^{\ast }$-algebras that are stably isomorphic to AT algebras;

11. separable simple $C^{\ast }$-algebras that are stably isomorphic to sequential direct limits of one dimensional NCCW complexes;

12. separable $C^{\ast }$-algebras with strict comparison of positive elements.

In particular, when $\alpha :G\to Aut\left(A\right)$ is an action of a finite group $G$ on $A$ with the Rokhlin property in the sense of Nawata, the properties $\left(1\right)$–$\left(12\right)$ are inherited to the fixed point algebra $A^{\alpha }$ and the crossed product algebra $A{⋊}_{\alpha }G$ from $A$. In the case of a finite index inclusion of unital $C^{\ast }$-algebras $P\subset A$ if the conditional expectation $E:A\to P$ has the Rokhlin property in the sense of Izumi, the previous results except (11) are observed in previous works.

Using the direct method, we prove the Hyers–Ulam stability of permuting triderivations and permuting trihomomorphisms in Banach algebras and unital $C^{\ast }$-algebras associated with the triadditive $s$-functional inequality $\parallel f\left(x+y,z-w,a+b\right)+f\left(x-y,z+w,a-b\right)-2f\left(x,z,a\right)+2f\left(x,w,b\right)-2f\left(y,z,b\right)+2f\left(y,w,a\right)\parallel \le \parallel s\left(2f\left(\left(x+y\right)∕2,z-w,a+b\right)+2f\left(\left(x-y\right)∕2,z+w,a-b\right)-2f\left(x,z,a\right)+2f\left(x,w,b\right)-2f\left(y,z,b\right)+2f\left(y,w,a\right)\right)\parallel$, where $s$ is a fixed nonzero complex number with $\left|s\right|<1$.

Let $\mathrm{\Gamma }$ be a finite group. Consider the wreath product $G_n:={\mathrm{\Gamma }}^n⋊S_n$ and the subgroup $K_n:={\mathrm{\Delta }}_n\times S_n\subseteq G_n$, where $S_n$ is the symmetric group and ${\mathrm{\Delta }}_n$ is the diagonal subgroup of ${\mathrm{\Gamma }}^n$. For certain values of $n$ (which depend on the group $\mathrm{\Gamma }$), the pair $\left(G_n,K_n\right)$ is a Gelfand pair. It is not known for all finite groups which values of $n$ result in Gelfand pairs. Building off the work of Benson–Ratcliff [4], we obtain a result which simplifies the computation of multiplicities of irreducible representations in certain tensor product representations, then apply this result to show that for $\mathrm{\Gamma }=S_k$, $k\ge 5$, $\left(G_n,K_n\right)$ is a Gelfand pair exactly when $n=1,2$.

Let $p$ and $q$ be multiplicatively independent integers. We show that the complex group ring of $\mathbb{Z}\left[1∕pq\right]⋊{\mathbb{Z}}^2$ admits a unique $C^{\ast }$-norm. The proof uses a characterization, due to Furstenberg, of closed $\times p$- and $\times q$-invariant subsets of $\mathbb{𝕋}$.

We give an analytical and topological proof of the uniqueness of the ground state of the nonlinear Schrödinger equation defined on the Hyperbolic space ${ℍ}^d$ when the power type nonlinearity has $H^1\left({ℍ}^d\right)$-subcritical exponent ($1<p<1+4∕\left(d-2\right)$ for $d\ge 3$ and $1<p<+\infty$ for $d=2$) and the phase $\lambda$ is positive. Differently from what it is available in the literature, we use the polar model of ${ℍ}^d$ and we do not take advantage of the dual Euclidean problem. Our proof of uniqueness uses the shooting method, some new monotonicity formulas and the geometry of the potential energy.

We studied boundary value problem of fourth-order impulsive differential equations involving oscillatory nonlinear term. Infinitely many nonnegative, distinct, classical solutions are obtained by using minimum value theory. Moreover, the proof that the weak solution is the classic solution is supplemented.

We say that a triangle $\mathrm{\Delta }$ tiles the polygon $\mathcal{𝒫}$ if $\mathcal{𝒫}$ can be decomposed into finitely many nonoverlapping triangles similar to $\mathrm{\Delta }$. Let $\mathcal{ℛ}$ be a regular pentagon. In this paper we prove that a right triangle tiles $\mathcal{ℛ}$ if and only if this triangle has an acute angle $\frac{\pi }{5}$.

Necessary and sufficient conditions for the Cauchy mean value to be Schur-$m$ power convex or concave are given. As applications, Schur-$m$ power convexity of the exponential mean is discussed, and a comparative inequality between the Gini mean and the Stolarsky mean is given.

We first establish a Hilbert-type inequality and its equivalent Hardy form with best possible constant factors by constructing a homogeneous kernel function involving hyperbolic functions. Furthermore, we introduce Bernoulli numbers and the partial fraction expansions of trigonometric functions, and then we present several special and interesting Hilbert-type inequalities, in which the constant factors are represented by Bernoulli numbers and by some higher derivatives of trigonometric functions.

We investigate the fractional $p$-Laplacian equation ${\left(-\mathrm{\Delta }\right)}_p^{s}u=f\left(x,u,\nabla u\right)$. We obtain the monotonicity and symmetry of positive solutions of the fractional $p$-Laplacian equation on bounded and unbounded domains. Specially, for unbounded case, we present a new decay at infinity.

This paper is mainly concerned with existence of entire positive radial large solutions for a class of Monge–Ampère type equations:

$$det{D}^{2}u\left(x\right)-\alpha \mathrm{\Delta }u=a\left(\left|x\right|\right)f\left(u\right),x\in {ℝ}^N,$$

and systems:

$$\begin{array}{ccccc}\hfill det{D}^{2}u\left(x\right)-\alpha \mathrm{\Delta }u& =a\left(\left|x\right|\right)f\left(v\right),\hfill & \hfill & x\in {ℝ}^N,\hfill & \hfill \\ \hfill det{D}^{2}v\left(x\right)-\beta \mathrm{\Delta }v& =b\left(\left|x\right|\right)g\left(u\right),\hfill & \hfill & x\in {ℝ}^N,\hfill \end{array}$$

where $det{D}^{2}u$ is the so-called Monge–Ampère operator, $\mathrm{\Delta }$ is the classical Laplace operator, $N\ge 2$, $\alpha ,\beta$ are positive constants, $f,g:\left[0,\infty \right)\to \left[0,\infty \right)$ are continuous and nondecreasing, and $a,b:{ℝ}^N\to \left[0,\infty \right)$ are continuous.