We will describe the topological type of the discriminant curve of the morphism \((\ell , f)\), where \(\ell \) is a smooth curve and f is an irreducible curve (branch) of multiplicity less than five or a branch such that the difference between its Milnor number and Tjurina number is less than 3. We prove that for a branch of these families, the topological type of the discriminant curve is determined

For a given set \(S\subseteq \mathbb {Z}_m\) and \(\overline{n}\in \mathbb {Z}_m\), \(R_S(\overline{n})\) is defined as the number of solutions of the equation \(\overline{n}=\overline{s}+\overline{s'}\) with unordered pair \((\overline{s},\overline{s'})\in S^2\) and \(\overline{s}\ne \overline{s'}\). In this paper, we prove that if \(m=2^id,\,i\ge 1,\,2\not \mid d, d > 1\) then there exist two sets

We examine the ideals of nowhere dense sets in three topologies on the set of positive integers, namely Furstenberg’s, Rizza’s and the common division topology. We mainly concentrate on inclusions between these ideals, we present a diagram showing these and we explore all possible inclusions between them. We present a formula for the closure of a set in the common division topology. We answer a question

We consider semigroups strongly Morita equivalent to a fixed monoid. We prove that such semigroups are precisely the enlargements of that monoid. We also show that enlargements of a given group are precisely the Rees matrix semigroups over that group.

We present a tight parametrical Hermite–Hadamard type inequality with probability measure, which yields a considerably closer upper bound for the mean value of convex function than the classical one. Our inequality becomes equality not only with affine functions, but also with a family of V-shaped curves determined by the parameter. The residual (a distance between two sides) of this inequality is

The purpose of this paper is to give, through the second degree character, new characterizations of a part of the family of quasi-symmetric forms. In fact, thanks to the Stieltjes function and also the moments, we give necessary and sufficient conditions for a regular form to be at the same time of the second degree, quasi-symmetric and semiclassical one of class two. We focus our attention not only

Let \(n\ne 0\) be an integer. A set of m distinct positive integers \(\{a_1,a_2,\ldots ,a_m\}\) is called a D(n)-m-tuple if \(a_ia_j + n\) is a perfect square for all \(1\le i < j \le m\). Let k be a positive integer. In this paper, we prove that if \(\{k,k+1,c,d\}\) is a \(D(-k)\)-quadruple with \(c>1\), then \(d=1\). The proof relies not only on standard methods in this field (Baker’s linear forms

We introduce sequential warped product submanifolds of Kaehler manifolds, provide examples and establish Chen’s inequality for such submanifolds. The equality case is also studied. Moreover, inspired by Lawson and Simons’s integral currrent’s theorem on a submanifold, we find a similar pinching inequality for a sequential warped product submanifold and obtain geometric results when the equality case

Let G be an additive finite abelian group. For a sequence T over G and \(g\in G\), let \(\mathrm {v}_{g}(T)\) denote the multiplicity of g in T. Let \(\mathcal {B}(G)\) denote the set of all zero-sum sequences over G. For \(\Omega \subset \mathcal {B}(G)\), let \(\mathsf {d}_{\Omega }(G)\) be the smallest integer t such that every sequence S over G of length \(|S|\ge t\) has a subsequence in \(\Omega

In this paper we present the well-defined solution of the following system of higher-order rational difference equations: $$\begin{aligned} x^{(j)}_{n+1}=\frac{x^{(j+1)\pmod {p}}_{n-k}}{a+bx^{(j+1)\pmod {p}}_{n-k}},\quad n, p, k\in \mathbb {N}_0 , j=\overline{1,p}, \end{aligned}$$ where the parameters a, b are nonzero real numbers and the initial values \(x^{(j)}_{-k}\), \(x^{(j)}_{-k+1}\),\(\ldots

For any positive integer m, let \(\mathbb {Z}_{m}\) be the set of residue classes modulo m. For \(A\subseteq \mathbb {Z}_{m}\) and \(\overline{n}\in \mathbb {Z}_{m}\), let representation function \(R_{A}(\overline{n})\) denote the number of solutions of the equation \(\overline{n}=\overline{a}+\overline{a'}\) with ordered pairs \((\overline{a}, \overline{a'})\in A \times A\). In this paper, we determine

We provide a unique normal form for rank two irregular connections on the Riemann sphere. In fact, we provide a birational model where we introduce apparent singular points and where the bundle has a fixed Birkhoff–Grothendieck decomposition. The essential poles and the apparent poles provide two parabolic structures. The first one only depends on the formal type of the singular points. The latter

In this paper, we study how the roots of the Kac polynomials \(W_n(z) = \sum _{k=0}^{n-1} \xi _k z^k\) concentrate around the unit circle when the coefficients of \(W_n\) are independent and identically distributed nondegenerate real random variables. It is well known that the roots of a Kac polynomial concentrate around the unit circle as \(n\rightarrow \infty \) if and only if \({\mathbb {E}}[\log

Lagrange introduced the notion of Schwarzian derivative and Thurston discovered its mysterious properties playing a role similar to that of curvature on Riemannian manifolds. Here we continue our studies on the development of the Schwarzian derivative on Finsler manifolds. First, we obtain an integrability condition for the Möbius equations. Then we obtain a rigidity result as follows; Let (M, F) be

We study the Galvin property. We show that various square principles imply that the cofinality of the Galvin number is uncountable (or even greater than \(\aleph _1\)). We prove that the proper forcing axiom is consistent with a strong negation of the Glavin property.

For any integer l and any positive integer n, let \( \sigma _{l}(n)=\sum _{d\mid n}d^{l}\). In 1936, Erdős proved that the set of positive integers n with \(\sigma _1 (n+1)\ge \sigma _1 (n)\) has natural density \(\frac{1}{2}\). Recently, M. Kobayashi and T. Trudgian showed that the set of positive integers n with \(\sigma _1 (2n+1)\ge \sigma _1 (2n)\) has natural density between 0.053 and 0.055. In

A ring R is called a left FGF ring if every finitely generated left R-module can be embedded in a free left R-module. It is proved that a group ring RG is left FGF if and only if R is left FGF and G is a finite group.

In this paper, by constructing some new q-identities, we prove some q-congruences. For example, for any odd integer \(n>1\), we show that $$\begin{aligned} \sum _{k=0}^{n-1}\frac{(q^{-1};q^2)_k}{(q;q)_k}q^k\equiv & {} (-1)^{(n+1)/2}q^{(n^2-1)/4}-(1+q)[n]\pmod {\Phi _n(q)^2},\\ \sum _{k=0}^{n-1}\frac{(q^3;q^2)_k}{(q;q)_k}q^k\equiv & {} (-1)^{(n+1)/2}q^{(n^2-9)/4}+\frac{1+q}{q^2}[n]\pmod {\Phi _n(q)^2}

In this paper, the authors have obtained \(L_1\)-approximations of functions f in \( {{\,\mathrm{Lip}\,}}(\alpha ,1) \) \( (0 < \alpha \le 1) \) by trigonometrical polynomials \( N_n (f;x)\) whenever the nonnegative and nonincreasing sequence \( (p_n )\) satisfies certain conditions. This enables the authors to approximate \( f \in {{\,\mathrm{Lip}\,}}(\alpha ,p) \) \((0< \alpha \le 1,1\le p < \infty

The aim of this paper is to continue the work started in Pavlović (Filomat 30(14):3725–3731, 2016) . We investigate further the properties of the local closure function and the spaces defined by it using common ideals, like ideals of finite sets, countable sets, closed and discrete sets, scattered sets and nowhere dense sets. Also, closure compatibility between the topology and the ideal, idempotency

Let \(\mathcal {A}\) and \(\mathcal {B}\) be abelian categories and \({\mathbf {F}} :\mathcal {A}\rightarrow \mathcal {B}\) an additive and right exact functor which is perfect, and let \(({\mathbf {F}},\mathcal {B})\) be the left comma category. We give an equivalent characterization of Gorenstein projective objects in \(({\mathbf {F}},\mathcal {B})\) in terms of Gorenstein projective objects in \(\mathcal

Estimation of the tail index of heavy-tailed distributions and its applications are essential in many research areas. We propose a class of weighted least squares (WLS) estimators for the Parzen tail index. Our approach is based on the method developed by Holan and McElroy (J Stat Plan Inference 140(12):3693–3708, 2010). We investigate consistency and asymptotic normality of the WLS estimators. Through

This article concerns the existence and multiplicity of solutions for a p(x)-Kirchhoff-type systems with a nonstandard growth condition. By a direct variational approach exponent Sobolev spaces, under growth conditions on the reaction terms, we establish the existence and multiplicity of solutions.

For any fixed integers a and b greater than 1, we study the Diophantine equation \(a^x+(ab+1)^y=b^z\). First, we describe a heuristic list of the positive integer solutions x, y and z of the equation. Finally, we solve the equation in some particular cases, which supports the validity of our list of solutions.

Spectral theory has become a central area of studies on commutative hypergroups. Spectral analysis and synthesis have been proved for different classes of hypergroups. In their paper (Fechner and Székelyhidi in Ann Univ Sci Budapest Sect Comput), the authors introduced and studied an infinite join of general finite hypergroups. Here we prove that under some general conditions spectral analysis and

In this paper, we first prove the existence and uniqueness of the solutions for a delayed hyperbolic partial differential equation by applying the progressive contraction technique introduced by Burton (Nonlinear Dyn Syst Theory 16(4): 366–371, 2016; Fixed Point Theory 20(1): 107–113, 2019) to the corresponding fixed-point problem. Then we derive a Hyers–Ulam stability result for this differential

In this note, an upper bound for the sum of fractional parts of certain smooth functions is given. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due to the use of Weyl’s bound for exponential sums and a device used by Popov.

Let \(p\equiv -q \equiv 5\pmod 8\) be two prime integers. In this paper, we investigate the unit groups of the fields \( L_1 =\mathbb {Q}(\sqrt{2}, \sqrt{p}, \sqrt{q}, \sqrt{-1} )\) and \( L_1^+=\mathbb {Q}(\sqrt{2}, \sqrt{p}, \sqrt{q} )\). Furthermore , we give the second 2-class groups of the subextensions of \(L_1\) as well as the 2-class groups of the fields \( L_n =\mathbb {Q}( \sqrt{p}, \sqrt{q}

Let \(\alpha = (1+\sqrt{5})/2\) and define the lower and upper Wythoff sequences by \(a_i = \lfloor i \alpha \rfloor \), \(b_i = \lfloor i \alpha ^2 \rfloor \) for \(i \ge 1\). In a recent interesting paper, Kawsumarng et al. proved a number of results about numbers representable as sums of the form \(a_i + a_j\), \(b_i + b_j\), \(a_i + b_j\), and so forth. In this paper I show how to derive all of

Recently in Dikranjan et al. (Fund Math 249: 185–209, 2020) an uncountable Borel subgroup \(t^s_{(2^n)}({\mathbb T}) \) (called statistically characterized subgroup) was constructed containing the Prüfer group \({\mathbb Z}(2^\infty )\) using the notion of statistical convergence. This note is based on the recent work (Bose et al. in Acta Math Hungar, 2020) which helps us to show that an uncountable

For a strictly increasing sequence \(A=\{a_i\}_{i=1}^{\infty }\) of positive integers, Borwein (Can Math Bull 20:117–118, 1978) proved that \(\sum _{i=1}^{n} { {{\,\mathrm{lcm}\,}}(a_i,a_{i+1})}^{-1}\le 1-\frac{1}{2^n}\) for any positive integer n, where the equality holds if and only if \(a_i=2^{i-1}\) for \(i=1,2, \ldots ,n+1\). Let r be an integer with \(3\le r\le 7\), Qian (C R Acad Sci Paris Ser

We show that the monodromy of Klassen’s genus two open book for \(P^2 \times S^1\) is the Y-homeomorphism of Lickorish, which is also known as the crosscap slide. Similarly, we show that \(S^2 \mathbin {\widetilde{\times }}S^1\) admits a genus two open book whose monodromy is the crosscap transposition. Moreover, we show that each of \(P^2 \times S^1\) and \(S^2 \mathbin {\widetilde{\times }}S^1\)

We establish the geometrical bearing on Legendrian submanifolds of Sasakian space forms in terms of r-almost Newton–Ricci solitons (r-anrs) with the potential function \(\psi : M^{n} \rightarrow \mathcal {R}\). Also, we discuss the Legendrian immersion of Ricci solitons and obtain conditions for L-minimal and totally geodesic under Newton transformation. Finally, we furnish our paper with the study

We prove that the largest convex shape that can be placed inside a given convex shape \(Q \subset \mathbb {R}^{d}\) in any desired orientation is the largest inscribed ball of Q. The statement is true both when “largest” means “largest volume” and when it means “largest surface area”. The ball is the unique solution, except when maximizing the perimeter in the two-dimensional case.

In this paper we prove that the time part of the germ of an analytic family of vector fields with a Hopf bifurcation is rigid in the parameter. Time parts are associated with the temporal invariant of the analytic classification. Because the eigenvalues at zero are complex conjugate, time parts usually unfold in the hyperbolic direction, where the singular points are linearizable. We first identify

We present algebraic and geometric classifications of the 4-dimensional complex nilpotent right alternative algebras. Specifically, we find that, up to isomorphism, there are only 9 non-isomorphic nontrivial nilpotent right alternative algebras. The corresponding geometric variety has dimension 13 and it is determined by the Zariski closure of 4 rigid algebras and one one-parametric family of algebras

For an integer \(k\ge 2\), let \((L_n^{(k)})_n\) be the k-generalized Lucas sequence which starts with \(0,\ldots ,0,2,1\) (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all k-generalized Lucas numbers which are Fibonacci, Pell or Pell–Lucas numbers, i.e., we study the Diophantine equations \(L_n^{(k)}=F_m\), \(L^{(k)}_n = P_m\) and \(L_n^{(k)}=Q_m\)

Consider the classical problem of rational simultaneous approximation to a point in \({\mathbb {R}}^{n}\). The optimal lower bound on the gap between the induced ordinary and uniform approximation exponents has been established by Marnat and Moshchevitin in 2018. Recently Nguyen, Poels and Roy provided information on the best approximating rational vectors to the points where the gap is close to this

For the Fibonacci sequence the identity \(F_n^2+F_{n+1}^2 = F_{2n+1}\) holds for all \(n \ge 0\). Let \({\mathcal {X}}:= (X_\ell )_{\ell \ge 1}\) be the sequence of X-coordinates of the positive integer solutions (X, Y) of the Pell equation \(X^2-dY^2=\pm 1\) corresponding to a nonsquare integer \(d>1\). In this paper, we investigate all positive nonsquare integers d for which there are at least two

In this paper we describe \({\mathcal {R}}\)-unipotent semigroups being regular extensions of a left regular band by an \(\mathcal {R}\)-unipotent semigroup T as certain subsemigroups of a wreath product of a left regular band by T. We obtain Szendrei’s result that each E-unitary \({\mathcal {R}}\)-unipotent semigroup is embeddable into a semidirect product of a left regular band by a group. Further

In this paper two new tensor fields on real hypersurfaces in complex quadric are introduced. Real hypersurfaces on which the derivatives of the tensor fields with respect to the Levi-Civita connection and the k-th generalized Tanaka–Webster connection of them coincide are studied leading to new classification results on real hypersurfaces in a complex quadric.

We consider finite colourings of finite products \(X_1\times X_2\times \cdot \cdot \cdot \times X_n\) of infinite sets and determine what is the minimal number of colours a subproduct \(Y_1\times Y_2\times \cdot \cdot \cdot \times Y_n\) of infinite subsets could achieve.

For each integer \(n\ge 1\) we consider the unique polynomials \(P, Q\in {\mathbb {Q}}[x]\) of smallest degree n that are solutions of the equation \(P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1\). We derive numerous properties of these polynomials and their derivatives, including explicit expansions, differential equations, recurrence relations, generating functions, resultants, discriminants, and irreducibility

In this paper, we define the localization operator associated with the Riemann–Liouville operator, and show that it is not only bounded, but it is also in the Schatten–von Neumann class. We also give a trace formula when the symbol function is nonnegative.

Let d(n) be the divisor function and denote by [t] the integral part of the real number t. In this short note, we prove that \(\sum _{n\le x} \ d\Big (\Big [\frac{x}{n}\Big ]\Big ) = x\sum _{m\ge 1}\frac{\ d(m)}{m(m+1)} + O_{\varepsilon }\big (x^{11/23+\varepsilon }\big )\) for \(x\rightarrow \infty \).

In this note we confirm two conjectural supercongruences of double sums of binomial coefficients due to El Bachraoui.

The goal of this paper is to give some theorems which relate to the problem of classifying combinatorial (resp. smooth) closed 5-manifolds up to piecewise-linear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edge-colored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced

The main aim of this paper is to provide some range-type criteria for the essentially self-adjointness of symmetric linear relations in real or complex Hilbert spaces. The main tool is a matrix whose entries are certain linear relations.

We consider the partitions of n into parts not congruent to 0, \(\pm 3\pmod {12}\) and provide some new results related to the number of these partitions. In this context, we derive a new parity result involving sums of partition numbers p(n) and squares in arithmetic progressions: for \(n\geqslant 0\), $$\begin{aligned} \sum _{8k+1\text { square}} p\big (n-3k\big ) \equiv 1 \pmod {2} \end{aligned}$$

Let K be a field and put $${\mathcal {A}}:=\{(i,j,k,m)\in \mathbb {N}^{4}:\;i\le j\;\text{ and }\;m\le k\}$$ . For any given $$A\in {\mathcal {A}}$$ we consider the sequence of polynomials $$(r_{A,n}(x))_{n\in \mathbb {N}}$$ defined by the recurrence $$\begin{aligned} r_{A,n}(x)=f_{n}(x)r_{A,n-1}(x)-v_{n}x^{m}r_{A,n-2}(x),\;n\ge 2, \end{aligned}$$ where the initial polynomials $$r_{A,0}, r_{A,1}\in

In this paper we prove a natural generalization of Gerzon’s bound. Our bound improves the Delsarte, Goethals and Seidel’s upper bound in a special case. Our proof is a simple application of the linear algebra bound method.

In this paper, we establish a necessary and sufficient criterion for a finite metabelian 2-group G whose abelianized $$G^{ab}$$ is of type $$(2, 2^m)$$ , with $$m\ge 2$$ , to be metacyclic. This criterion is based on the rank of the maximal subgroup of G which contains the three normal subgroups of G of index 4. Then, we apply this result to study the structure of the Galois group of the maximal unramified

We establish direct estimates of the rate of weighted simultaneous approximation by the Szasz–Mirakjan operator for smooth functions in the supremum norm on the non-negative semi-axis. We consider Jacobi-type weights. The estimates are stated in terms of appropriate moduli of smoothness or K-functionals.

In this paper, we give some sufficient conditions for the Dominguez–Lorenzo condition in terms of the James type constant $$J_{X,t}(\tau )$$ , the coefficient of weak orthogonality $$\mu (X)$$ and the Dominguez Benavides coefficient R(1, X), which imply the existence of fixed points for multivalued nonexpansive mappings. Our results extend some well known results in the recent literature.

A new generalization of Grassmannians in supergeometry, called $$\nu $$ -Grassmannians, are constructed by gluing $$\nu $$ -domains. By a $$\nu $$ -domain, we mean a superdomain with an odd involution say $$\nu $$ on its structure sheaf, as morphism of modules. Then we show that $$\nu $$ -Grassmannians are homogeneous superspaces. In addition, in the last section, a supergroup associated to the odd

In this note we construct an algorithm generating any discrete distribution with an arbitrary coin (and, as a result, with arbitrary initial distribution). The coin need not be fair and the target distribution can be supported on a countable set.

Let $$\Lambda (n)$$ be the von Mangoldt function, and let [t] be the integral part of real number t. In this note we prove that the asymptotic formula $$\begin{aligned} \sum _{n\leqslant x} \Lambda \Big (\Big [\frac{x}{n}\Big ]\Big ) = x\sum _{d\geqslant 1}\frac{\Lambda (d)}{d(d+1)} + O_{\varepsilon }\big (x^{35/71+\varepsilon }\big ) \end{aligned}$$ holds as $$x\rightarrow \infty $$ for any $$\varepsilon

It is well known that for each $$n\ge 0$$ , there is a continuous map $$ f :S^{n}\rightarrow \partial \bigtriangleup ^{n+1}$$ with the disjoint support property. Since $$S^{n}$$ and $$\partial \bigtriangleup ^{n+1}$$ are homeomorphic, it is natural to ask whether or not there is a homeomorphism $$ h :S^{n}\rightarrow \partial \bigtriangleup ^{n+1}$$ with the disjoint support property. In this paper

Suppose $P$ is a symmetric convex polygon in the plane. We give a polynomial time algorithm that decides if $P$ can tile the plane by transations at some level (not necessarily at level one; this is multiple tiling). The main technical contribution is a polynomial time algorithm that selects, if this is possible, for each $j=1,2,\ldots,n$ one of two given vectors $e_j$ or $\tau_j$ so that the selection

We study the algebraic independence of Laurent series in positive characteristic which can be fast approximated by rational functions. This can be seen as a completion of the results obtained by Chaichana and Laohakosol (Period Math Hung 55(1):35–59, 2007).