Expository notes on the Schwarz lemma born out of some lectures given on the subject.

Axiomatizing mathematical structures and theories, or postulating them as Russell (1919) put it, is a goal of mathematical logic. Some axiomatic systems are mere definitions, such as the axioms of Group Theory; but some are much deeper, such as the axioms of complete ordered fields with which real analysis starts. Groups abound in the mathematical sciences, while by Dedekind’s theorem (1888) there exists only one complete ordered field, up to isomorphism. Cayley’s theorem (1854) in abstract algebra implies that the axioms of group theory completely axiomatize the class of permutation sets that are closed under composition and inversion.

In this expository article, we survey some old and new results on the first-order axiomatizability of various mathematical structures. As we will see, axiomatizability of some structures are still unsolved questions in mathematics, and several results have been open problems in the past. We will also review identities over addition, multiplication, and exponentiation that hold in the set of positive real numbers; and will have a look at Tarski’s high school problem (1969) and its solution.

For $\alpha \in ℝ$, the general Randić index of a graph $G$ is defined as $R_{\alpha }(G)={\sum }_{uv\in E(G)}{[d(u)d(v)]}^{\alpha }$, where $E(G)$ is the edge set of $G$, and $d(u)$ and $d(v)$ are the degrees of vertices $u$ and $v$ in $G$, respectively. We give a survey containing known extremal results about the general Randić index. We present results for general graphs, trees, unicyclic graphs, bicyclic graphs and chemical graphs.

Bohr phenomenon for analytic functions $f$, where $f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$, was first introduced by Harald Bohr in $1914$ and deals with finding the largest radius $r_f$, , such that the inequality holds for $\left|z\right|=r\le r_f$ whenever holds in the unit disk . The Bohr phenomenon for harmonic functions of the form $f(z)=h+ḡ$, where $h(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ and $g(z)={\sum }_{n=1}^{\infty }{b}_n{z}^n$, is to find the largest radius $r_f$, such that

$$\sum _{n=1}^{\infty }(|a_n|+|b_n|)|z{|}^n\le d(f(0),\partial f(\mathbb{𝔻}))$$

holds for $\left|z\right|\le r_f$, where $d(f(0),\partial f(\mathbb{𝔻}))$ is the Euclidean distance between $f(0)$ and the boundary of $f(\mathbb{𝔻})$. We prove several improved versions of the sharp Bohr radius for the classes of harmonic and univalent functions. Further, we prove several corollaries as a consequence of the main results.

This paper is concerned with the existence and uniqueness of solutions for a coupled system of Caputo-type sequential fractional differential equations supplemented with boundary conditions. The existence of solutions is derived by applying the Leray–Schauder alternative, while the uniqueness of solutions is established via Banach’s contraction principle. Moreover, some necessary conditions for the Hyers–Ulam-type stability to the solutions of the boundary value problem are developed. Finally the results are supported by example.

We study Hyers–Ulam stability for linear differential equations in the sense of quaternion-valued framework. This shows that Laplace transformation is also valid for finding the approximate solution for linear quaternion-valued differential equations.

We describe the entire solutions for two kinds of nonlinear differential-difference equations of the form

$$f^n(z)+\omega f^{n-1}(z)f^{\prime }(z)+q_1(z)e^{Q_1(z)}{f}_{c_1}+q_2(z)e^{Q_2(z)}{f}_{c_2}=u(z)e^{v(z)},n\ge 3$$

and

$$f^n(z)+q(z)e^{Q(z)}f(z+c)=u(z)e^{v(z)},n\ge 2,$$

where $q,Q,u,v,q_j(z),Q_j(z)$, for $j=1,2$, are polynomials such that $Q(z)$ and at least one of $Q_j(z)$ are not constants, $q(z)$ and $q_j(z)$ are not identically zero, and $\omega ,c_1,c_2,c$ are constants. Our results improve and generalize some previous results.

The two-dimensional Aw–Rascle model can describe the traffic flow with lane changing. We are interested in the four-wave Riemann problem for the two-dimensional Aw–Rascle model with Chaplygin pressures, which is fully linearly degenerate. The contact discontinuity and $\delta$-shock wave may develop in solutions. We are concerned with the solutions only involving contact discontinuities. As a result, twelve kinds of structures are derived.

Using methods from the theory of uniform algebras, we give a simple proof of an approximation result of Sahutoğlu and Tikaradze with $L^{\infty }$-pseudoconvex domains replaced by the open sets for which Gleason’s problem is solvable.

For prime power $p^n$, let $\mathrm{\Theta }(k,p^n)$ denote the minimal $s$ such that for any integers $a_i$ coprime to $p$ the congruence

$$a_1{x}_1^k+a_2{x}_2^k+\cdots +a_s{x}_s^k\equiv 0(\text{mod}{p}^n)$$

has a primitive solution ($p\nmid x_i$ for some $i$), and ${\mathrm{\Theta }}^{\ast }(k,p^n)$ the minimal $s$ such that there is always a solution with $p\nmid x_i$ for all $i$, $1\le i\le s$. We obtain a number of estimates for these two quantities when the group $A$ of $k$-th powers mod $p$ has at least two elements, including, $\mathrm{\Theta }(k,p)\ll {\text{log}}^{3}k$, $\mathrm{\Theta }(k,p)\ll {\text{log}}^{5∕2}k$ if $\left|A\right|$ is composite, $\mathrm{\Theta }(k,p^n)\ll k^{\sqrt{5}∕\sqrt{{\text{ log}}_{2}k}}{\text{ log}}^{2}k,$ ${\mathrm{\Theta }}^{\ast }(k,p^n)\le 8k^{5∕{\text{log }}_2(t_1∕C)}$ for some absolute constant $C$, where , and ${\mathrm{\Theta }}^{\ast }(k,p)\le 8$ for .

The Clifford spectrum is an elegant way to define the joint spectrum of several Hermitian operators. Examples as small as a triple of $2\times 2$ matrices can have a Clifford spectrum that is a two-dimensional manifold. To date, there are very few noncommuting examples where the Clifford spectrum has been computed.

Our main goal is the generation of various noncommuting examples of three or four Hermitian matrices where we can calculate the Clifford spectrum. In some cases, our results are rigorous, perhaps involving assistance from a computer algebra package. In other cases, we rely on computer estimation to plot the spectrum. We make a few conjectures, for example, regarding the possible spaces that can appear as Clifford spectra.

Inspired by classical puzzles in geometry that ask about probabilities of geometric phenomena, we give an explicit formula for the probability that a random triangle on a flat torus is homotopically trivial. Our main tool for this computation involves reducing the problem to a new invariant of measurable sets in the plane that is unchanged under area-preserving affine transformations. Our result show that this probability is minimized at all rectangular tori and maximized at the regular hexagonal torus.

We show that, under certain conditions, restricted and biased exponential sums and Walsh transforms of symmetric and rotation symmetric Boolean functions are, as in the case of nonbiased domain, $C$-finite sequences. We also prove that under other conditions, these sequences are $P$-finite, which is a somewhat different behavior than their nonbiased counterparts. We further show that exponential sums and Walsh transforms of a family of rotation symmetric monomials over the restricted domain $E_{n,j}=\{x\in {\mathbb{𝔽}}_2^n:wt(x)=j\}$ ($wt(x)$ is the weight of the vector $x$) are given by polynomials of degree at most $j$, and so, they are also $C$-finite sequences. Finally, we also present a study of the behavior of symmetric Boolean functions under these biased transforms.

Suppose that ${\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2\in ℝ$. Let be irrational and of finite type such that $1,{\alpha }_1^{-1},{\alpha }_2^{-1}$ are linearly independent over $\mathbb{Q}$. Let $c$ be a real number in the range . In this paper, it is proved that there exist infinitely many primes in the intersection of the Beatty sequences ${\mathcal{ℬ}}_{{\alpha }_1,{\beta }_1}=\lfloor {\alpha }_{1}n+{\beta }_1\rfloor$, ${\mathcal{ℬ}}_{{\alpha }_2,{\beta }_2}=\lfloor {\alpha }_{2}n+{\beta }_2\rfloor$ and the Piatetski-Shapiro sequence ${\mathcal{𝒩}}^{(c)}=\lfloor n^c\rfloor$. Moreover, we also give a sketch of the proof of Piatetski-Shapiro primes in the intersection of multiple Beatty sequences.

To study statistical properties of modular forms, including for instance Sato–Tate like problems, it is essential to be able to compute a large number of Fourier coefficients. We show that this can be achieved in level 4 for a large range of half-integral weights by making use of one of three explicit bases, the elements of which can be calculated via fast power series operations.

We systematically study the notion of cross ratios and energy pairings on metric graphs and electrical networks. We show that several foundational results on electrical networks and metric graphs immediately follow from the basic properties of cross ratios. For example, the projection matrices of Kirchhoff have natural (and efficiently computable) expressions in terms of cross ratios. We prove a generalized version of Rayleigh’s law, relating energy pairings and cross ratios on metric graphs before and after contracting an edge segment. Quantitative versions of Rayleigh’s law for effective resistances, potential kernels, and cross ratios will follow as immediate corollaries.

We show that the derivative of a log-analytic function is log-analytic. We prove that log-analytic functions exhibit strong quasianalytic properties. We establish the parametric version of Tamm’s theorem for log-analytic functions.

We prove the Lipschitz regularity of the Kobayashi and Carathéodory metrics on suitable domains.

We deal with the oscillation of a class of generalized fractional integrodifferential equations containing a forcing term. We establish sufficient conditions to obtain some oscillation criteria for the Riemann–Liouville and Caputo cases. Moreover, we provide several examples to support our results.

We give the sharp and refined bounds for the Tsallis relative operator entropy by using the improved Hermite–Hadamard inequality. As an application of our result, we give new bounds for the generalized relative operator entropy. We refine the bounds of the relative operator entropy announced by Fujii and Kamei and provide the precise and sharp bounds.

We investigate the existence of multiple solutions of boundary value problems for higher order impulsive equations. By using variational and perturbative methods, we show the existence of infinitely many solutions in the nonsymmetric case.

We describe many explicit constructions of Halphen pencils of index two and their associated rational elliptic surfaces. For each of the types of singular fiber that occur we construct at least one Halphen pencil yielding that type of fiber and in fact, for some, we construct all possible examples. More generally, we also study the singularities of a curve ${\mathcal{𝒞}}_{3m}$ appearing in a Halphen pencil of index $m$ in terms of the log canonical threshold of the pair $({\mathbb{P}}^2,{\mathcal{𝒞}}_{3m})$.