Recently, Araya, Celikbas, Sadeghi, and Takahashi proved a theorem about the vanishing of self extensions of finitely generated modules over commutative Noetherian rings. The aim of this paper is to obtain extensions of their result over algebras.

We characterize spheres as the unique complete properly immersed self-shrinkers in arbitrary codimension satisfying a geometric inequality.

We consider the Fano scheme \(F_k(X)\) of k-dimensional linear subspaces contained in a complete intersection \(X \subset {\mathbb {P}}^n\) of multi-degree \({\underline{d}} = (d_1, \ldots , d_s)\). Our main result is an extension of a result of Riedl and Yang concerning Fano schemes of lines on very general hypersurfaces: we consider the case when X is a very general complete intersection and \(\Pi

We investigate the class number one problem for a parametric family of real quadratic fields of the form \(\mathbb {Q}( \sqrt{m^2+4r})\) for certain positive integers m and r.

In this paper, we use the Fourier–Hermite functionals to extend the structure of the Cameron–Martin translation theorem on an abstract Wiener space \((H,B,\nu )\). The directional function in our translation theorem may not be in the Cameron–Martin subspace H of B. We then proceed to obtain an explicit formula for our general translation theorem.

The cover-avoidance property and its variations have become central in the study of finite groups. Along with this widespread attention comes a need to construct examples and counterexamples. The main goal of this work is to apply the theory of direct products to systematically facilitate such constructions.

In this paper, we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset A of \(\mathbb {Z}_n{\setminus } \{0\}\) of size k such that \(\sum _{z\in A} z\not = 0\), it is possible to find an ordering \((a_1,\ldots ,a_k)\) of the elements of A such that the partial sums \(s_i=\sum _{j=1}^i a_j\), \(i=1,\ldots ,k\), are nonzero and

For arbitrary \(S^{1}\)-actions on \(S^{m}_{{\mathbb {Q}}}\), \(S^{n}_{{\mathbb {Q}}}\), and \(S^{m}_{{\mathbb {Q}}}\times S^{n}_{{\mathbb {Q}}}\), we show the conditions for the tenability of the homotopy equivalence \((S^{m}_{{\mathbb {Q}}})^{hS^{1}}\times (S^{n}_{{\mathbb {Q}}})^{hS^{1}}\simeq (S^{m}_{{\mathbb {Q}}}\times S^{n}_{{\mathbb {Q}}})^{hS^{1}}\). Here, \(X^{hS^1}\) denotes the homotopy

Using the sub-supersolution method, we study the existence of positive solutions for the anisotropic problem $$\begin{aligned} -\sum _{i=1}^N\frac{\partial }{\partial x_i}\left( \left| \frac{\partial u}{\partial x_i}\right| ^{p_i-2}\frac{\partial u}{\partial x_i}\right) =\lambda u^{q-1} \end{aligned}$$(0.1) where \(\Omega \) is a bounded and regular domain of \({\mathbb {R}}^N\), \(q>1\), and \(\lambda

Let G be a finite soluble group and \(G^{(k)}\) the kth term of the derived series of G. We prove that \(G^{(k)}\) is nilpotent if and only if \(|ab|=|a||b|\) for any \(\delta _k\)-values \(a,b\in G\) of coprime orders. In the course of the proof, we establish the following result of independent interest: let P be a Sylow p-subgroup of G. Then \(P\cap G^{(k)}\) is generated by \(\delta _k\)-values

In this short note, studying 3-dimensional compact and minimal submanifolds of the \((3+p)\)-dimensional unit sphere \({\mathbb {S}}^{3+p}(1)\), we establish two rigidity theorems in terms of the Ricci curvature. The first theorem related to hypersurfaces of \({\mathbb {S}}^4(1)\) gives a new characterization of the minimal Clifford torus, whereas the second theorem is about the Legendrian submanifolds

Let \({\mathcal {C}}({\mathcal {H}})={\mathcal {B}}({\mathcal {H}})/{\mathcal {K}}({\mathcal {H}})\) be the Calkin algebra (\({\mathcal {B}}({\mathcal {H}})\) the algebra of bounded operators on the Hilbert space \({\mathcal {H}}\), \({\mathcal {K}}({\mathcal {H}})\) the ideal of compact operators, and \(\pi :{\mathcal {B}}({\mathcal {H}})\rightarrow {\mathcal {C}}({\mathcal {H}})\) the quotient map)

Using the recent work of Bettiol, we show that a first-order conformal deformation of Wilking’s metric of almost-positive sectional curvature on \(S^2\times S^3\) yields a family of metrics with strictly positive average of sectional curvatures of any pair of 2-planes that are separated by a minimal distance in the 2-Grassmanian. A result of Smale allows us to conclude that every closed simply connected

Let p be a prime number, and let \({K}\) be a number field. For \(p=2\), assume moreover that \({K}\) is totally imaginary. In this note, we prove the existence of asymptotically good extensions \({L}/{K}\) of cohomological dimension 2 in which infinitely many primes split completely. Our result is inspired by a recent work of Hajir, Maire, and Ramakrishna.

We investigate the inverse q-numerical range of \(2\times 2\) matrices. A quartic curve is formulated that generates unit vectors for the inverse q-numerical range according to Tsing’s circle formula. Properties of this quartic curve are discussed and corresponding examples are described.

Let k be an algebraically closed field of characteristic 0 and H a non-semisimple monomial Hopf algebra. Extending a previous result of Kassel (Ann Math Blaise Pascal 20: 175–191, 2013), we prove that some (not necessarily bi-)Galois objects over H are determined up to H-comodule algebra isomorphism by their polynomial H-identities.

In this note, faithfully flat descent for projectivity is generalized to pure descent for projectivity.

Let w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that \(N_w(1)\ge |G|^{k-1}\), where for \(g\in G\), the quantity \(N_w(g)\) is the number of k-tuples \((g_1,\ldots ,g_k)\in G^{(k)}\) such that \(w(g_1,\ldots ,g_k)={g}\). Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit’s conjecture

It is well known that the Riemann zeta function, as well as several other L-functions, is universal in the strip \(1/2<\sigma <1\); this is certainly not true for \(\sigma >1\). Answering a question of Bombieri and Ghosh, we give a simple characterization of the analytic functions approximable by translates of L-functions in the half-plane of absolute convergence. Actually, this is a special case of

On the Wasserstein space over a complete, separable, non-compact, locally compact length space, we consider the horo-functions associated to sequences of atomic measures. We show the existence of co-rays for any prescribed initial probability measure with respect to a sequence of atomic measures and show that co-rays are negative gradient curves in some sense. Some other fundamental results of this

Congruence modular and congruence distributive varieties can be characterized by the existence of sequences of Gumm and Jónsson terms, respectively. Such sequences have variable lengths, in general. It is immediate from the above paragraph that there is a variety with Gumm terms but without Jónsson terms. We prove the unexpected result that, on the other hand, if some variety has both kinds of terms

Let k be a field of characteristic 0. Using the method of idealization, we show that there is a non-Koszul, quadratic, Artinian, Gorenstein, standard graded k-algebra of regularity 3 and codimension 8, answering a question of Mastroeni, Schenck, and Stillman. We also show that this example is minimal in the sense that no other idealization that is non-Koszul, quadratic, Artinian, Gorenstein algebra

Let C be a smooth curve of degree \(d_0\) lying on a smooth surface in projective space. Exploiting a criterion due to I. Reider we study when the divisors in a linear series of degree \(d \le d_0\) on C are contained in hyperplanes. For a K3 surface S with Picard group \({\mathbb {Z}}^2\) we succeed if S contains a line.

During half a century, singular traces on ideals of Hilbert space operators have been constructed by looking for linear forms on associated sequence ideals. Only recently, the author was able to eliminate this auxiliary step by directly applying Banach’s version of the extension theorem; see (Integral Equ. Oper. Theory 91, 21, 2019 and 92, 7, 2020). Of course, the relationship between the new approach

Let G be a 2-generated group. The generating graph \(\Gamma (G)\) of G is the graph whose vertices are the elements of G and where two vertices g and h are adjacent if \(G = \langle g, h \rangle .\) This definition can be extended to a 2-generated profinite group G, considering in this case topological generation. We prove that the set V(G) of non-isolated vertices of \(\Gamma (G)\) is closed in G

We prove that generalized loop spaces of Hartogs manifolds are Hilbert–Hartogs. We prove also that Hilbert–Hartogs manifolds possess a better extension properties than it is postulated in their definition. Finally, we give a list of examples of Hilbert–Hartogs manifolds.

Let \(S={\textsf {k}}[X_1,\dots , X_n]\) be a polynomial ring, where \({\textsf {k}}\) is a field. This article deals with the defining ideal of the Rees algebra of a squarefree monomial ideal generated in degree \(n-2\). As a consequence, we prove that Betti numbers of powers of the cover ideal of the complement graph of a tree do not depend on the choice of the tree. Further, we study the regularity

We prove that, up to adding a complement, every modular representation of a finite group admits a finite resolution by permutation modules.

We use the notion of the optimal plan associated with the Fuglede p-modulus of a family of Borel measures to derive formulas for the p-modulus and the extremal function in many special cases. Among others, we deduce Rodin’s formula for a family of Hausdorff measures associated with leaves of a foliation defined by a single chart.

A connected Lie group admitting an expansive automorphism is known to be nilpotent, but not all nilpotent Lie groups admit expansive automorphisms. In this article, we find sufficient conditions for a class of nilpotent Lie groups to admit expansive automorphisms.

We show that the center set of reversible cubic systems, close to the symmetric Hamiltonian system \(x'=y, y'= x-x^3\), has two irreducible components of co-dimension two in the parameter space. One of them corresponds to the Hamiltonian stratum, the other to systems which are a polynomial pull back of an appropriate linear system.

A subgroup H of a group G is said to be complemented in G if there exists a subgroup K of G such that \(G=HK\) and \(H \cap K=1\). We prove that, for a locally soluble group G, all cyclic subgroups are complemented if and only if it is the semidirect product of groups \(A= {{\,\mathrm{Dr}\,}}_{i \in I} A_i\) by \(B={{\,\mathrm{Dr}\,}}_{j \in J} B_j\), where all factors \(A_i\) and \(B_j\) are finite

We discuss certain geometric properties for area stationary currents and currents with integrable mean curvature, so called “enclosure theorems”. As a consequence, we obtain non-existence results for currents with connected support. Finally, we extend these results to currents in submanifolds and state a non-existence result for stationary currents in spheres.

Let \((R,{\mathfrak {m}},k)\) be a Noetherian local ring of dimension \(d\ge 4\). Assume that \(2\le i \le d-2\) is an integer and \(x_1,\ldots ,x_i\) is a part of a system of parameters for R. Let \(\Upsilon _i\) denote the set of all prime ideals \({\mathfrak {p}}\) of R such that \(\dim R/{\mathfrak {p}}=i+1\), \({\text {Supp}}H^i_{(x_1,\ldots ,x_i)R}(R/{\mathfrak {p}})=\{{\mathfrak {m}}\}\), and

We conjectured in Malle and Navarro (J Algebra 370:402–406, 2012) that a Sylow p-subgroup P of a finite group G is normal if and only if whenever p does not divide the multiplicity of \(\chi \in {{\text {Irr}}}(G)\) in the permutation character \((1_P)^G\), then p does not divide the degree \(\chi (1)\). In this note, we prove an analogue of this for p-Brauer characters.

We consider the control problem of the heat equation on bounded and unbounded domains, and more generally the corresponding inhomogeneous equation for the Schrödinger semigroup. We show that if the sequence of null-controls associated to an exhaustion of an unbounded domain converges, then the solutions do in the same way, and that the control cost estimate carries over to the limiting problem on the

We consider Mergelyan sets and Farrell sets for \(H^p\)\((1\le p < \infty )\) spaces in the unit disc for both the weak topology and the norm topology, and give a short proof of a theorem of Pérez-González which answers a question proposed by Rubel and Stray (J Approx Theory 37:44–50, 1983).

An open problem about finite geometric progressions in syndetic sets leads to a family of diophantine equations related to the commutativity of translation and multiplication by squares.

We consider the Cauchy problem for the dissipative nonlinear Schrödinger equation with a cubic nonlinear term \(\lambda |u|^2u\), where \(\lambda \in {\mathbb {C}}\) with Im \(\lambda < 0\). We prove the global existence of a unique solution and obtain the uniform estimate in the Gevrey class. Using the uniform regularity estimate, we show the \(L^2\)-decay rate for the solution which has the Gevrey

The heart of a cotorsion pair, which is a generalization of the heart of a t-structure, has been proven to be abelian on triangulated, exact, and extriangulated categories. On the other hand, the heart of a twin cotorsion pair is not always abelian (it is only semi-abelian). In this article, we study a special kind of hearts of twin cotorsion pairs induced by d-cluster tilting subcategories. We will

We define Rankin–Selberg series on general linear groups and study its growth rate by counting lattice points.

It is well-known that the side length of a regular hexagon is half the length of its longest diagonals. From this property, one can easily see that for every positive integer \(m>1\), any regular 6m-gon contains two non-congruent diagonals that are commensurable. In this paper, we show that if n is not a multiple of 6, then all pairs of diagonals of different lengths of a regular n-gon are incommensurable

Let A be a positive bounded linear operator acting on a complex Hilbert space \({\mathcal {H}}\). Our aim in this paper is to prove some A-numerical radius inequalities of bounded linear operators acting on \({\mathcal {H}}\) when an additional semi-inner product structure induced by A is considered. In particular, an alternative proof of a recent result proved in Moslehian et al. (Linear Algebra Appl

From the work of Erdős and Rényi from 1963, it is known that almost all graphs have no symmetry. In 2017, Lupini, Mančinska, and Roberson proved a quantum counterpart: Almost all graphs have no quantum symmetry. Here, the notion of quantum symmetry is phrased in terms of Banica’s definition of quantum automorphism groups of finite graphs from 2005, in the framework of Woronowicz’s compact quantum groups

In this short note, we obtain an integral inequality for closed Riemannian manifolds with positive scalar curvature and give some rigidity characterization of the equality case, which generalizes the recent results of Catino which deal with the conformally flat case, and of Huang and Ma which deal with the harmonic curvature case. Moreover, we obtain an integral pinching condition with non-negative

In this paper, we give various finiteness results concerning solutions of generalized Pell equations representable as sums of S-units with a fixed number of terms. In case of one term, our result is effective, while in case of more terms, we are able to bound the number of solutions.

Recently, the authors have established \(L^p\)-boundedness of vector-valued q-variational inequalities for averaging operators which take values in the Banach space satisfying the martingale cotype q property in Hong and Ma (Math Z 286(1–2):89–120, 2017). In this paper, we prove that the martingale cotype q property is also necessary for the vector-valued q-variational inequalities, which was a question

Let A be an Auslander algebra of global dimension equal to 2. We provide a necessary and sufficient condition for A to be a tilted algebra. In particular, A is a tilted algebra if and only if \({{\,\mathrm{pd}\,}}_{A}(\tau _{A}\Omega _{A}DA)\le 1\).

We find the exact best possible range of those \(p >1\) for which any \(\varphi \in A_1({\mathbb {R}})\), with \(A_1\) constant equal to c, must also belong to \(L^p\). In this way, we provide an alternative proof of the corresponding result in Bojarski and Sbordone (Studia Math 101(2):155–163, 1992) and Nikolidakis (Ann Acad Scient Fenn Math 40:949–955, 2015).

We study almost complex submanifolds of pseudo nearly Kähler manifolds. We show in particular that a 6 dimensional strict nearly Kähler manifold does not admit any 4 dimensional almost complex submanifolds. This generalises results obtained by Gray (Proc Am Math Soc 20:277–279, 1969) for the nearly Kähler 6-sphere and by Podestà and Spiro (J Geom Phys 60:156–164, 2010) in the Riemannian case.

We generalize results of Alladi, Dawsey, and Sweeting and Woo for Chebotarev densities to general densities of sets of primes. We show that if K is a number field and S is any set of prime ideals with natural density \(\delta (S)\) within the primes, then $$\begin{aligned} -\lim _{X \rightarrow \infty }\sum _{\begin{array}{c} 2 \le {\text {N}}(\mathfrak {a})\le X\\ \mathfrak {a} \in D(K,S) \end{array}}\frac{\mu

We generalize the recent work of Viazovska by constructing infinite families of Schwartz functions, suitable for Cohn–Elkies style linear programming bounds, using quasi-modular and modular forms. In particular, for dimensions \(d \equiv 0 \pmod {8}\), we give new constructions for obtaining sphere packing upper bounds via modular forms. In dimension 8 and 24, these exactly match the functions constructed

This note deals with the existence and uniqueness of a minimiser of the following Grötzsch type problem $$\begin{aligned} \inf _{f\in {\mathcal {F}}}\mathop {\iint }\limits _{Q_{1}}\psi ({\mathbb {K}}(z,f))dxdy \end{aligned}$$ under some conditions, where \({\mathcal {F}}\) denotes the set of all orientation preserving homeomorphisms f with the boundary correspondence. As an application, we consider

We consider the following nonlinear Schrödinger equation in \({\mathbb {R}}^N(N\ge 2)\): $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\lambda u=g(u), \\ u\in H^1({\mathbb {R}}^N), ~\mathop {\int }\nolimits _{{\mathbb {R}}^N} u^2=c, \end{array}\right. \end{aligned}$$ where \(c>0\) is a given constant, \(\lambda \in {\mathbb {R}}\) is a Lagrange multiplier, and \(g\in C^1({\mathbb {R}},{\mathbb

For a \(C^{\infty }\)-function f on \(\mathbb {R}\), \(S_n(f)\) denotes the n-th partial sum of the Taylor series of f with center at 0. Let \(f_1,f_2\) be two \(C^{\infty }\)-functions. We prove that even if the union \(\{S_n(f_1):n\in \mathbb {N}\}\cup \{S_n(f_2):n\in \mathbb {N}\}\) is dense in the space of continuous functions on \([-1,1]\) vanishing at 0 endowed with the topology of uniform convergence

In this paper, we will prove a Liouville theorem for poly-harmonic functions on \({{\mathbb {R}}}^{n}_{+}\) with Navier boundary conditions, that is, the nonnegative poly-harmonic functions u satisfying \(u(x)=o(|x|^{3})\) at \(\infty \) must assume the form $$\begin{aligned} u(x)=C x_{n} \end{aligned}$$ in \(\overline{{{\mathbb {R}}}^{n}_{+}}\), where \(n\ge 2\) and C is a nonnegative constant. The

Crofton’s formula of integral geometry evaluates the total motion invariant measure of the set of k-dimensional planes having nonempty intersection with a given convex body. This note deals with motion invariant measures on sets of pairs of hyperplanes or lines meeting a convex body. Particularly simple results are obtained if, and only if, the given body is of constant width in the first case, and

We study the periodic \(L_2\)-discrepancy of point sets in the d-dimensional torus. This discrepancy is intimately connected with the root-mean-square \(L_2\)-discrepancy of shifted point sets, with the notion of diaphony, and with the worst-case error of cubature formulas for the integration of periodic functions in Sobolev spaces of mixed smoothness. In discrepancy theory, many results are based

In Emamirad et al. (in: Semigroups of Operators - Theory and Applications, Springer, 2014; and Proc. Amer. Math. Soc. 140:2043–2052, 2012), the Black–Scholes semigroup was studied in various Banach spaces of continuous functions regarding chaotic behavior and the null volatility limit. Here we let the volatility and the interest rates be continuous functions on the half line \([0,\infty )\), and we