Szegő and Schweitzer proved the inequality

$\left(1\right)\sum _{k=1}^n\left(n+1-k\right)\left(n+2-k\right)ksinkx\ge 0$

for $n\in \mathbb{N};0\le x\le \frac{2\pi }{3}$.

We show that the following counterparts are valid:

$\left(2\right)\sum _{\genfrac{}{}{0ex}{}{k=1}{kodd}}^n\left(n+1-k\right)\left(n+2-k\right)ksinkx\ge 0$

for $n\in \mathbb{N};0\le x\le \pi$, and

$\left(3\right)\sum _{\genfrac{}{}{0ex}{}{k=1}{keven}}^n\left(n+1-k\right)\left(n+2-k\right)ksinkx\ge 0$

for $2\le n\in \mathbb{N};0\le x\le \frac{1}{2}arccos\left(-\frac{5}{6}\right)$.

The intervals $\left[0,\frac{2\pi }{3}\right]$, $\left[0,\pi \right]$ and $\left[0,\frac{1}{2}arccos\left(-\frac{5}{6}\right)\right]$ are the largest intervals of the form $\left[0,a\right]$ such that the inequalities (1), (2) and (3) are valid for all $n$. Moreover, we use (2) and (3) to obtain two one-parameter classes of absolutely monotonic functions and we show that these functions satisfy a certain functional inequality.

Gap balancing numbers are a certain generalization of balancing and cobalancing numbers that arise from studying the equation $T\left(L\right)+T\left(B\right)=T\left(m\right)$ where $T\left(i\right)$ is the $i$-th triangular number. In this paper, we survey early results, attempt to unify existing terminology, and extend prior findings. In addition, we further investigate the structure of classes of gap balancing numbers and related sequences, presenting new results and a conjecture regarding the number of classes based on the gap size.

This paper deals with various new formulas involving a quotient of two or more functions using applicable techniques of the differential transform method (DTM). These newly developed formulas could be helpful for solving delay differential equations and integro-differential equations, which are highly challenging to solve directly. Moreover, a delay differential equation is completely solved using various algorithms in MATLAB incorporating these new formulas.

BiHom-Novikov superalgebras are generalized Novikov superalgebras endowed with two commuting multiplicative linear maps. The main purpose of this paper is to show the structure and representation of BiHom-Novikov superalgebras. We obtain the relationship between Novikov superalgebras and BiHom-Novikov superalgebras. We show that quadratic BiHom-Novikov superalgebras are in connection with 2 different quadratic BiHom-Lie superalgebras. Moreover, we develop the 1-parameter formal deformation theory of BiHom-Novikov superalgebras.

We find all Weierstrass points on some of the bielliptic intermediate modular curves $X_{\mathrm{\Delta }}\left(N\right)$ whose bielliptic involutions are of Atkin–Lehner type.

Frobenius is generally considered the creator of character theory of finite groups. This achievement came from the study of the group determinant, which is the determinant of a matrix coming from the regular representation. We generalize several of Frobenius’ results about the group determinant and use them to find a new formulation of supercharacter theory in terms of factorizations of the group determinant.

We completely solve the Diophantine equation $x^2+2^{k}11^{\ell }19^m=y^n$ in integers $x,y\ge 1$; $k,\ell ,m\ge 0$ and $n\ge 3$ with $gcd\left(x,y\right)=1$, except the case $2|k$, $2\nmid \ell m$ and $5|n$. We use this result to recover some earlier results in the same direction.

We completely solve a system of operator equations, whose findings are utilized to generalize the notion of slant Toeplitz operators defined on the Lebesgue space of $n$-dimensional torus ${\mathbb{𝕋}}^n$. Ultimately, certain algebraic and spectral properties of the considered operators are investigated.

We give a complete and self-contained proof of a folklore theorem which says that in an Alexandrov space the distance between a point $\gamma \left(t\right)$ on a geodesic $\gamma$ and a compact set $K$ is a right-differentiable function of $t$. Moreover, the value of this right-derivative is given by the negative cosine of the minimal angle between the geodesic and any shortest path to the compact set (Theorem 4.3). Our treatment serves as a general introduction to metric geometry and relies only on the basic elements, such as comparison triangles and upper angles.

We study the practical exponential stability of nonlinear systems with generalized sufficient conditions. We use Lyapunov’s direct method, hence we extend previous results on the perturbed systems and the time-varying cascade systems. The last part is devoted to the study of the problem of practical $\alpha$-exponential stabilization for certain classes of nonlinear systems.

We use Krasnoselskii’s fixed point theorem to show that the neutral nonlinear summation-difference system with infinite delay $\mathrm{\Delta }x\left(n\right)=P\left(n\right)+A\left(n\right)x\left(n-\tau \left(n\right)\right)+\mathrm{\Delta }Q\left(n,x\left(n-g\left(n\right)\right)\right)+{\sum }_{k=-\infty }^{n}D\left(n,k\right)f\left(x\left(k\right)\right)$ has a periodic solution. We also use the contraction mapping principle to show that the periodic solution is unique. An example is given to illustrate our results.

We investigate analytic properties for some Euler products. In particular, we construct an Euler product ${\prod }_{p:\text{ prime}}{\left(1-{\alpha }_p{p}^{-s}\right)}^{-1}$$\left(\left|{\alpha }_p\right|=1\text{ for any prime number }p\right)$ with natural boundary on $\text{ Re}\left(s\right)=1$.

We study the compactness of finite sums of products of two Toeplitz operators on Hardy–Sobolev spaces over the unit polydisk $H_{\beta }^2\left({\mathbb{𝔻}}^n\right)$. We calculate the essential norm of these operators and answer the question of when a Toeplitz operator on $H_{\beta }^2\left({\mathbb{𝔻}}^n\right)$ is Fredholm.

Let $K=\mathbb{Q}\left(𝜃\right)$ be an algebraic number field with $𝜃$ satisfying an irreducible polynomial $x^6-m$ having integer coefficient $m$. We explicitly provide an integral basis of $K$ along with some examples. As an application, we show that when $m$ is a squarefree integer such that $m\equiv 2,3mod4$ and $m\not\equiv \pm 1mod9$, then $K$ has a power integral basis.

Let $c_r\left(n\right)$ be the number of representations of a positive integer $n$ as a sum of $r$ squares, where representations with different orders and different signs are counted as distinct. We prove a combinatorial identity relating this quantity to the sum of inverses of odd divisors of $n$:

$\sum _{oddd|n}\frac{1}{d}=\frac{1}{2}\sum _{r=1}^n\frac{{\left(-1\right)}^{n+r}}{r}\left(\genfrac{}{}{0.0pt}{}{n}{r}\right)c_r\left(n\right).$

Given a module $X$ over a ring $\mathrm{\Lambda }$ its stability class consists of all modules $X^{\prime }$ such that $X\oplus {\mathrm{\Lambda }}^a\cong X^{\prime }\oplus {\mathrm{\Lambda }}^b$ for some positive integers $a$, $b$. If the ring $\mathrm{\Lambda }$ is weakly finite then the stability class of a finitely generated $\mathrm{\Lambda }$-module has the structure of a tree. We show that if, in addition, $X$ is a generalized torsion module its stability class has the same shape as that of the zero module. In consequence we construct examples of nonprojective modules whose stability classes have arbitrarily large amounts of branching.

Our aim is to prove the following result. Let $X$ be a real or complex Banach space, let $\mathcal{ℒ}\left(X\right)$ be the algebra of all bounded linear operators on $X$ and let $\mathcal{𝒜}\left(X\right)\subseteq \mathcal{ℒ}\left(X\right)$ be a standard operator algebra, which possesses the identity operator. Suppose there exists a linear mapping $F:\mathcal{𝒜}\left(X\right)\to \mathcal{ℒ}\left(X\right)$ satisfying the relation $F\left(A^{n+2}\right)={\sum }_{i=1}^{n+1}{A}^{i-1}F\left(A^2\right)A^{n+1-i}-{\sum }_{i=1}^n{A}^{i}F\left(A\right)A^{n+1-i}$ for all $A\in \mathcal{𝒜}\left(X\right)$ and some fixed integer $n\ge 1$. In this case $F$ is of the form $F\left(A\right)=A{B}_1+B_{2}A$ for all $A\in \mathcal{𝒜}\left(X\right)$ and some fixed $B_1,B_2\in \mathcal{ℒ}\left(X\right)$. In particular, $F$ is continuous.

Let $\mathcal{𝒜}$ be a unital $\ast$-algebra with the unit $I$. Assume that $\mathcal{𝒜}$ contains a nontrivial projection $P$ which satisfies $X\mathcal{𝒜}P=0$ implies $X=0$ and $X\mathcal{𝒜}\left(I-P\right)=0$ implies $X=0$. In this paper, it is shown that $\mathrm{\Phi }$ is a nonlinear $\ast$-Jordan-type derivation on $\mathcal{𝒜}$ if and only if $\mathrm{\Phi }$ is an additive $\ast$-derivation. As applications, the nonlinear $\ast$-Jordan-type derivations on prime $\ast$-algebras, von Neumann algebras with no central summands of type $I_1$, factor von Neumann algebras and standard operator algebras are characterized.

We describe the Jacobson topology of the primitive ideal space of self-similar $k$-graph $C^{\ast }$-algebras under certain conditions.

We study the well-posedness of the 5-th Korteweg–de Vries (KdV) equation in the space $H^{2+s}\left(0,r\right)$ when the initial data is drawn from $H^{2+s}\left(0,r\right)$ and the boundary data $\left(h_1\left(t\right),h_2\left(t\right),\dots ,h_5\left(t\right)\right)$ lies in the product space $H^{s_1}\left(0,T\right),\dots ,H^{s_5}\left(0,T\right)$ for some appropriate indices $s_1,s_2,\dots ,s_5$ that depend on $s$. As we will see later, the natural choices of $s_1,s_2,\dots ,s_5$ are $s_1=\left(s+2\right)∕5$, $s_2=\left(s+1\right)∕5$, $s_3=s∕5$, $s_4=\left(s-1\right)∕5$, $s_5=\left(s-2\right)∕5$. We first use contraction mapping method to obtain the local solution of the IBVP, furthermore to get the global solution by the a prior estimate.

A class of third-order nonlinear autonomous differential equations is analyzed and solved. The first integral method suitable for the third-order autonomous differential equation is introduced and the first integrals of the equation are obtained by this method. Besides this, the functional transformations and Lie group theory are used to solve the equation.

We shall prove that if $X$ is a Hausdorff paracompactum with at least one nontrivial quasicomponent, then none of its Čech systems can be commutative and none can support the structure of an approximate inverse system.

We study the equations $m∕n=1∕x\pm 1∕y\pm 1∕z$, where $m=4,5,6,7$ and is a given positive integer. Except for the case $m∕n=1∕x+1∕y+1∕z$ (i.e., all signs are positive), which are known conjectures of Erdős–Straus ($m=4$), Sierpiński ($m=5$) and Schinzel ($m\ge 1$), we show that the equations (with at least one minus sign) have solutions $x,y,z$, where for sufficiently large $n$.

We propose and prove some new generalized Gronwall-type inequalities with respect to an appropriate function $\psi$. We consider the integral inequalities including not only one integral but also two integrals with doubly singular kernels. We apply these inequalities to study the existence and stability of solution of fractional differential equations with weakly singular sources.

Ternary paths consist of an upstep of one unit, a downstep of two units, never go below the $x$-axis, and return to the $x$-axis. This paper addresses the enumeration of partial ternary paths, ending at a given level $i$, reading the path either from left-to-right or from right-to-left. Since the paths are not symmetric with respect to left vs. right, as classical Dyck paths, this leads to different results. The right-to-left enumeration is quite challenging, but leads at the end to very satisfying results. The methods are elementary (solving systems of linear equations). In this way, several conjectures left open in Naiomi Cameron’s Ph.D. thesis could be successfully settled.

We construct special conic configurations from some point configurations which are the singularities of the dual of a quartic curve.

We employ the self-improving property (backward propagation) for the discrete Muckenhoupt class ${\mathcal{𝒜}}_p$, to prove that both discrete Hardy and discrete Hardy–Littlewood maximal operators are bounded on the usual weighted Lebesgue space ${\ell }_u^p\left({\mathbb{Z}}^{+}\right)$ if and only if the weight $u$ belongs to ${\mathcal{𝒜}}_p$. Some weak boundedness results for the Hardy–Littlewood maximal operators will also be discussed. To the best of the authors’ knowledge, the results are essentially new and have not been discussed before.

The main goal of this paper is to study the Poisson universal enveloping algebra of a generalized Poisson–Ore extension. We prove such a Poisson universal enveloping algebra is a twisted tensor product.