We settle J. Wetzel’s 1970’s conjecture and show that a \(30^{\circ }\) circular sector of unit radius can accommodate every planar arc of unit length. Leo Moser asked in 1966 for the (convex) region with the smallest area in the plane that can accommodate each arc of unit length. With area \(\pi /12,\) this sector is the smallest such set presently known. Moser’s question has prompted a multitude

In this paper, we prove that every Diophantine quadruple in \(\mathbb {Z}[i][X]\) is regular. More precisely, we prove that if \(\{a, b, c, d\}\) is a set of four non-zero polynomials from \(\mathbb {Z}[i][X]\), not all constant, such that the product of any two of its distinct elements increased by 1 is a square of a polynomial from \(\mathbb {Z}[i][X]\), then $$\begin{aligned} (a+b-c-d)^2=4(ab+1)(cd+1)

In this paper, a local version of the topological pressure of dynamical systems is presented. It is a function defined on the product space which does not depend on any measure. It is shown that, for any invariant measure, integration of the introduced function with respect to its corresponding diagonal measure results in the metric pressure of the dynamical system.

We prove that the inequality $$\begin{aligned} \Gamma (x+1)\le \frac{x^2+\beta }{x+\beta } \end{aligned}$$ holds for all \(x\in [0,1]\), \(\beta \ge {\beta ^{*}}\), with the best possible constant $$\begin{aligned} \beta ^*=\max _{0.1\le x\le 0.3} f(x)=1.75527\ldots , \end{aligned}$$ where f is given by $$\begin{aligned} f(x)=\frac{x\Gamma (x+1)-x^2}{1-\Gamma (x+1)}. \end{aligned}$$ This refines bounds

Studying eigenvalues of square matrices is a traditional and fundamental direction in linear algebra. Quaternion matrices constitute an important and extensively useful subclass of square matrices. In the paper, the authors (1) introduce the concept of “generalized eigenvalues of quaternion matrix”; (2) give some properties of generalized eigenvalues for a regular quaternion matrix pair; (3) establish

In this paper we introduce a new conformable derivative call it mixed conformable partial derivative, which obeys classical properties, including linearity, product rule, quotient rule and vanishing derivatives for constant functions. As an application, we establish an Opial type inequality for the mixed second order conformable partial derivatives.

A real valued function f defined on a real open interval I is called \(\Phi \)-monotone if, for all \(x,y\in I\) with \(x\le y\) it satisfies $$\begin{aligned} f(x)\le f(y)+\Phi (y-x), \end{aligned}$$ where \( \Phi :[0,\ell (I) [ \rightarrow \mathbb {R}_+\) is a given nonnegative error function, where \(\ell (I)\) denotes the length of the interval I. If f and \(-f\) are simultaneously \(\Phi \)-monotone

Let \(\mathbb {N}\) be the set of nonnegative integers. For any set \(A \subset \mathbb {N}\), let \(R_1(A, n)\), \(R_2(A, n)\) and \(R_3(A, n)\) be the number of representations of n as \(n=a+a',a, a'\in A\); \(n=a+a',a, a'\in A\), \(a

In this paper, we consider plane algebraic curves of genus zero defined over number fields having three simple points at infinity such that their minimal fields of definition are totally real, and we determine an upper bound of polynomial type for the size of their integral points.

For a given integer \(k\ge 3\), a sequence A of nonnegative integers is called an \(AP_k\)-covering sequence if there exists an integer \(n_0\) such that, if \(n>n_0\), then there exist \(a_1\in A\), \(\ldots \), \(a_{k-1}\in A\), \(a_1

In this work, we study non-characteristic domains of Heisenberg groups. We prove that bounded domains which are diffeomorphic to the solid torus having the center of the group as rotation axis, are non-characteristic. Then we state the following conjecture: the bounded non-characteristic domains of the Heisenberg group of dimension 1 are diffeomorphic to a solid torus having the center of the group

Let \(\mathbb N\) be the set of positive integers, and denote by $$\begin{aligned} \lambda (A)=\inf \{t>0:\sum _{a\in A} a^{-t}<\infty \} \end{aligned}$$ the convergence exponent of \(A\subset \mathbb N\). For \(0

Let \(P_k\) denote the kth term of the Pell sequence. In this paper we find all solutions of the exponential Diophantine equation \(P_n+P_m = y^s\) in positive integer variables (m, n, y, s) under the assumption \(n \equiv m \pmod 2\).

We explore the existence of irreducible and reducible arc-sections in an irreducible hypersurface singularity germ along finite projections. In particular we provide examples of irreducible isolated hypersurface singularities for which no irreducible arc-sections exist, and show that reducible ones always exist. Moreover, we give an algorithm to check if a given projection allows irreducible arc-sections

The aim of this article is to give a characterization of strongly quasipositive quasi-alternating links and detect new classes of strongly quasipositive Montesinos links and non-strongly quasipositive Montesinos links. In this direction, we show that, if L is an oriented quasi-alternating link with a quasi-alternating crossing c such that \(L_0\) is alternating (where \(L_0\) has the induced orientation)

Let \(\alpha = (1+\sqrt{5})/2\) be the golden ratio, and let \(B(\alpha ) = (\left\lfloor n\alpha \right\rfloor )_{n\ge 1}\) and \(B(\alpha ^2) = \left( \left\lfloor n\alpha ^2\right\rfloor \right) _{n\ge 1}\) be the lower and upper Wythoff sequences, respectively. In this article, we obtain a new estimate concerning the fractional part \(\{n\alpha \}\) and study the sumsets associated with Wythoff

Let \(a>1,b\) be two positive integers where the square-free part of b is 2pq with p, q two distinct odd primes. Recently, Cipu (Proc Am Math Soc 146:983–992, 2018) proved that if one of the following conditions holds: (i) \(2a^2-1\) is not a perfect square, (ii) \(\{p\pmod 8,q\pmod 8\}\ne \{1,3\}\), then the equations $$\begin{aligned} x^2-(a^2-1)y^2=1 \quad \text {and} \quad y^2-bz^2=1 \end{aligned}$$

We survey Ozsváth–Szabó’s bordered approach to knot Floer homology. After a quick introduction to knot Floer homology, we introduce the relevant algebraic concepts (\(\mathcal {A}_\infty \)-modules, type D-structures, box tensor product, etc.), we discuss partial Kauffman states, the construction of the boundary algebra, and sketch Ozsváth and Szabó’s analytic construction of the type D-structure associated

Recently, using a formula of Carlitz, the second author proved a q-congruence conjectured by Tauraso. In this note we utilize Carlitz’s formula to prove two more similar q-congruences. We also propose a conjectural q-congruence that refines one of our q-congruences.

Our paper begins with a revision of the spectral theory for commutative Banach algebras, which enables us to prove the \(L^p_{\omega }\)-conjecture for locally compact abelian groups. We follow an alternative approach to the one known in the literature. In particular, we do not resort to any structural theorems for locally compact groups at this stage. Subsequently, we discuss nilpotent, locally compact

Let (a, b, c) be pairwise relatively prime integers such that \(a^2 + b^2 = c^2\) . In 1956, Jeśmanowicz conjectured that the only solution of \(a^x + b^y = c^z\) in positive integers is \((x,y,z)=(2,2,2)\). In this note we prove a polynomial analogue of this conjecture.

This paper investigates the optimal dividend problem in a jump-diffusion risk model with debit interest. In this model, the insurer could borrow money at a debit interest when the surplus turns negative. However, when the negative surplus attains a certain critical level, the business stops and absolute ruin happens at this moment. A sufficient condition under which the optimal dividend strategy is

We will present proofs for two conjectures stated in Rot (Homotopy classes of proper maps out of vector bundles, 2020. arXiv:1808.08073). The first one is that for an arbitrary manifold W, the homotopy classes of proper maps \(W\times \mathbb {R}^n\rightarrow \mathbb {R}^{k+n}\) stabilise as \(n\rightarrow \infty \), and the second one is that in a stable range there is a Pontryagin–Thom type bijection

In this short note we use methods of Friedl, Livingston and Zentner to show that there are knots that are not algebraically concordant to a connected sum of positive and negative L-space knots.

The generic element of the moduli space of logarithmic connections with parabolic points on holomorphic vector bundle over the Riemann sphere can be represented by a Fuchsian equation with some singularities and some apparent singularities. We analyze the case of rank 3 vector bundle which leads to third order Fuchsian equation. We find coordinates on an open subset of the moduli space and we construct

In this paper we consider a link \(B_{n,\rho }\) between Baskakov type operators \(B_{n,\infty }\) and genuine Baskakov–Durrmeyer type operators \( B_{n,1}\) depending on a positive real parameter \(\rho \). The topic of the present paper is the pointwise limit relation \(\left( B_{n,\rho }f\right) \left( x\right) \rightarrow \left( B_{n,\infty }f\right) \left( x\right) \) as \(\rho \rightarrow \infty

We characterize p-rational real quadratic fields in terms of generalized Fibonacci numbers. We then use this characterization to give numerical evidence to a conjecture of Greenberg asserting the existence of p-rational multi-quadratic fields of arbitrary degree \(2^{t}\), \(t\ge 1\).

By using the Furstenberg–Ellis–Namioka structure theorem, we give a decomposition theorem for the Banach algebra \({\mathcal {M}}(G)\), i.e. the Banach algebra of those complex regular Borel measures on a compact Hausdorff admissible right topological (or simply CHART) group G for which the natural convolution product makes sense, generalizing an existing result due to Lau and Loy. Next, we characterize

In this paper we prove that if \(\{a,b,c\}\) is a Diophantine triple with \(a

Let \(k=k_0(\root 3 \of {d})\) be a cubic Kummer extension of \(k_0=\mathbb {Q}(\zeta _3)\) with \(d>1\) a cube-free integer and \(\zeta _3\) a primitive third root of unity. Denote by \(C_{k,3}^{(\sigma )}\) the 3-group of ambiguous classes of the extension \(k/k_0\) with relative group \(G={\text {Gal}}(k/k_0)=\langle \sigma \rangle \). The aims of this paper are to characterize all extensions \(k/k_0\)

A generalization of the well-known Fibonacci and Lucas sequences are the k-Fibonacci and k-Lucas sequences with some fixed integer \(k\ge 2\). For these sequences the first k terms are \(0,\ldots ,0,1\) and \(0,\ldots ,0,2,1\), respectively, and each term afterwards is the sum of the preceding k terms. Here we find all pairs of k-Fibonacci and k-Lucas numbers multiplicatively dependent.

Any sequence of d-dimensional boxes of edge length smaller than or equal to 1 with total volume not greater than \((3-2\sqrt{2})\cdot 3^{-d}\) can be packed online into the d-dimensional unit cube.

Let \(\Omega \) be an open, simply connected, bounded subset of the complex plane \(\mathbb {C}\) with a rectifiable boundary \({\partial {\Omega }}\). We investigate the relation between the Hardy–Smirnov space \(H^2(\Omega )\) and the closed span of the exponential system \(\{e^{nz}\}_{n=1}^{\infty }\) with respect to the Hardy–Smirnov norm \(\Vert \cdot \Vert _\Omega \). Depending on the “height”

The present paper deals with some particular properties of special series of Fourier coefficients of the class of functions with bounded variation with respect to general orthonormal systems (ONS). The obtained results demonstrate that the properties of the general ONS and of the classical ONSs (trigonometric, Haar, Walsh system) are different in some cases.

In this paper we generalize the result (obtained by the author earlier in the paper titled “Products and coproducts in the category \({{\mathcal {S}}}(B)\) of Segal topological algebras”) about the existence of the coproduct of objects of the category \({\mathcal S}(B)\) from finite collection of objects of \({{\mathcal {S}}}(B)\) to any collection of objects of \({{\mathcal {S}}}(B)\).

Let \((m,\ n)\) be fixed positive integers such that \(m>n,\ \gcd (m,\ n)=1\) and \( mn\equiv 0 \pmod 2\). Then the triple \((m^2-n^2,\ 2mn,\ m^2+n^2)\) is called a primitive Pythagorean triple. In 1956, Jeśmanowicz (Wiadom Math 1(2):196–202, 1955/1956 ) conjectured that the equation \((m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z\) has only the positive integer solution \((x,\ y,\ z)=(2,\ 2,\ 2)\). This problem

In this work, we generalize the von Neumann and Davis theorems on the extension of the convexity of a group-invariant norm (resp. function) from the subspace of diagonal matrices to the whole space of complex (resp. hermitian) matrices. Our generalizations go in two directions: (1) We replace the space of complex (hermitian) matrices and its subspace of diagonal matrices by an Eaton triple and its

First, we prove that the Diophantine system $$\begin{aligned} f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q) \end{aligned}$$ has infinitely many integer solutions for \(f(X)=X(X+a)\) with nonzero integers \(a\equiv 0,1,4\pmod {5}\). Second, we show that the above Diophantine system has an integer parametric solution for \(f(X)=X(X+a)\) with nonzero integers a, if there are integers m, n, k such that $$\begin{aligned}

In this paper, we give some fundamental equations of curvature tensors on Berwald submersions, which imply the relationship of flag curvatures of the associated manifolds and the fibers.

Maximal inequalities for the signed vector summands are established. Probabilistic estimations for the sets of appropriate signs are given. By use of the “transference technique” appropriate maximal inequalities are derived for the permutations. One application for orthogonal and Hadamard matrices is suggested.

The high-dimensional version of Fatou’s classical theorem asserts that the Poisson semigroup of a function \(f\in L_{p}(\mathbb {R}^{n}), \ 1\le p \le \infty \), converges to f non-tangentially at Lebesque points. In this paper we investigate the rate of non-tangential convergence of Poisson and metaharmonic semigroups at \(\mu \)-smoothness points of f.

In this paper we study q-analogues of Euler sums and present a new family of identities by using the method of Jackson q-integral representations of series. We then apply it to obtain a family of identities relating quadratic Euler sums to linear sums and q-polylogarithms. Furthermore, we use certain stuffle products to evaluate several q-series with q-harmonic numbers. Some interesting new results

We derive exact formulas for the \(\chi ^2\) statistics of the distribution of the leading digit of \(a^n\), where \(\log _{10} a\) has a large first or second partial quotient in its continued fraction expansion. We also give a mathematical explanation for the difference between the behavior of the discrepancy and that of the distribution of leading digits.

We count rational points of bounded height on the non-normal senary quartic hypersurface \(x^4=(y_1^2 + \cdots + y_4^2)z^2 \) in the spirit of Manin’s conjecture.

The purpose of the paper is to study Yamabe solitons on three-dimensional para-Sasakian, paracosymplectic and para-Kenmotsu manifolds. Mainly, we prove that the following: If the semi-Riemannian metric of a three-dimensional para-Sasakian manifold is a Yamabe soliton, then it is of constant scalar curvature, and the flow vector field V is Killing. In the next step, we prove that either the manifold

In the present work we deal with the quadratic decomposition of symmetric semiclassical polynomial sequences of class 2 orthogonal with respect to the positive definite weight \( | x^2-\frac{1}{2} |^p(1-x^2)^{-\frac{1}{2}}\), \( p > -1\), on \([-1,1]\). The coefficients of the three-term recurrence relation, the structure relation, the differential equation as well as some information about the zeros

First we show that there exist infinitely many distinct cyclic cubic number fields K such that the Fermat cubic \(x^3 + y^3 = z^3\) has non-trivial points in K. Second, we show that the Fermat quartic \(x^4 + y^4 = z^4\) can have no non-trivial points in any cyclic cubic number field. It remains an open question whether the Fermat quartic has any points in quartic number fields with Galois group of

The arithmetic Kakeya conjecture, formulated by Katz and Tao (Math Res Lett 6(5-6):625-630, 1999), is a statement about addition of finite sets. It is known to imply a form of the Kakeya conjecture, namely that the upper Minkowski dimension of a Besicovitch set in R n is n. In this note we discuss this conjecture, giving a number of equivalent forms of it. We show that a natural finite field variant

We study Diophantine equations of type f ( x ) = g ( y ) , where both f and g have at least two distinct critical points (roots of the derivative) and equal critical values at at most two distinct critical points. Various classical families of polynomials ( f n ) n are such that f n satisfies these assumptions for all n. Our results cover and generalize several results in the literature on the finiteness

In this work, we show that the modulus of smoothness in the periodic variable exponents Morrey spaces and the K-functional with respect to the periodic variable exponents Morrey spaces—the periodic variable exponents Sobolev–Morrey spaces are equivalent. As an application we use this characterization for obtaining an analogue of Whitney’s theorem in the spaces \({{\mathcal {M}}}_{p(\cdot ),\lambda