An extension of Wilf’s conjecture to affine semigroups

Let \(\mathcal {C}\subset \mathbb {Q}^p_+\) be a rational cone. An affine semigroup \(S\subset \mathcal {C}\) is a \(\mathcal {C}\)-semigroup whenever \((\mathcal {C}\setminus S)\cap \mathbb {N}^p\) has only a finite number of elements. In this work, we study the tree of \(\mathcal {C}\)-semigroups, give a method to generate it and study the \(\mathcal {C}\)-semigroups with minimal embedding dimension. We extend Wilf’s conjecture for numerical semigroups to \(\mathcal {C}\)-semigroups and give some families of \(\mathcal {C}\)-semigroups fulfilling the extended conjecture. Other conjectures formulated for numerical semigroups are also studied for \(\mathcal {C}\)-semigroups.     

Ladders and canonical varieties of completely regular semigroups

A completely regular semigroup is a (disjoint) union of its (maximal) subgroups. We consider it here with the unary operation of inversion within its maximal subgroups. Their totality \(\mathcal {C}\mathcal {R}\) forms a variety whose lattice of subvarieties is denoted by \(\mathcal {L}(\mathcal {C}\mathcal {R})\). On it, one defines the relations \(\mathbf {B}^\wedge \) and \(\mathbf {B}^\vee \) by

$$\begin{aligned} \begin{array}{lll} \mathcal {U}\ \mathbf {B}^\wedge \ \mathcal {V}&{} \Longleftrightarrow &{} \mathcal {U}\cap \mathcal {B} =\mathcal {V}\cap \mathcal {B}, \\ \mathcal {U}\ \mathbf {B}^\vee \ \mathcal {V}&{} \Longleftrightarrow &{} \mathcal {U}\vee \mathcal {B} =\mathcal {V}\vee \mathcal {B} , \end{array} \end{aligned}$$

respectively, where \(\mathcal {B}\) denotes the variety of all bands. This is a study of the interplay between the \(\cap \)-subsemilatice \(\triangle \) of \(\mathcal {L}(\mathcal {C}\mathcal {R})\) of upper ends of \(\mathbf {B}^\wedge \)-classes and their \(\mathbf {B}^\vee \)-classes. The main tool is the concept of a ladder and their \(\mathbf {B}^\vee \)-classes, an indispensable part of the important Polák’s theorem providing a construction for the join of varieties of completely regular semigroups. The paper includes the tables of ladders of the upper ends of most \(\mathbf {B}^\wedge \)-classes. Canonical varieties consist of two ascending countably infinite chains which generate most of the upper ends of \(\mathbf {B}^\wedge \)-classes.

     

A note on generalized skew derivations on Lie ideals

Let \(\mathcal {R}\) be a prime ring, \(\mathcal {Z(R)}\) its center, \(\mathcal {C}\) its extended centroid, \(\mathcal {L}\) a Lie ideal of \(\mathcal {R}, \mathcal {F}\) a generalized skew derivation associated with a skew derivation d and automorphism \(\alpha \). Assume that there exist \(t\ge 1\) and \(m,n\ge 0\) fixed integers such that \( vu = u^m\mathcal {F}(uv)^tu^n\) for all \(u,v \in \mathcal {L}\). Then it is shown that either \(\mathcal {L}\) is central or \(\mathrm{char}(\mathcal {R})=2, \mathcal {R}\subseteq \mathcal {M}_2(\mathcal {C})\), the ring of \(2\times 2\) matrices over \(\mathcal {C}, \mathcal {L}\) is commutative and \(u^2\in \mathcal {Z(R)}\), for all \(u\in \mathcal {L}\). In particular, if \(\mathcal {L}=[\mathcal {R,R}]\), then \(\mathcal {R}\) is commutative.     

Uniqueness of self-shrinkers to the degree-one curvature flow with a tangent cone at infinity

Given a smooth, symmetric and homogeneous of degree one function \(f\left( \lambda _{1},\ldots ,\lambda _{n}\right) \) satisfying \(\partial _{i}f>0\quad \forall \,i=1,\ldots , n\), and a properly embedded smooth cone \({\mathcal {C}}\) in \({\mathbb {R}}^{n+1}\), we show that under suitable conditions on f, there is at most one f self-shrinker (i.e. a hypersurface \(\Sigma \) in \({\mathbb {R}}^{n+1}\) satisfying \(f\left( \kappa _{1},\ldots ,\kappa _{n}\right) +\frac{1}{2}X\cdot N=0\), where \(\kappa _{1},\ldots ,\kappa _{n}\) are principal curvatures of \(\Sigma \)) that is asymptotic to the given cone \({\mathcal {C}}\) at infinity.     

Infra-nilmanifolds modeled on the group of uni-triangular matrices

Let \({\mathcal {N}}_m\) be the group of \(m\times m\) upper triangular real matrices with all the diagonal entries 1. Then it is an \((m-1)\)-step nilpotent Lie group, diffeomorphic to \({\mathbb {R}}^{\frac{1}{2} m(m-1)}\). It contains all the integer matrices as a lattice \(\Gamma _m\). The automorphism group of \({\mathcal {N}}_m \ (m\ge 4)\) turns out to be extremely small. In fact, \(\mathrm {Aut}({\mathcal {N}})=\mathcal {I} \rtimes \mathrm {Out}({\mathcal {N}})\), where \(\mathcal {I}\) is a connected, simply connected nilpotent Lie group, and \(\mathrm {Out}({\mathcal {N}})={{\tilde{K}}}={(\mathbb {R}^*)^{m-1}\rtimes \mathbb {Z}_2}\). With a nice left-invariant Riemannian metric on \({\mathcal {N}}\), the isometry group is \(\mathrm {Isom}({\mathcal {N}})= {\mathcal {N}} \rtimes K\), where \(K={(\mathbb {Z}_2)^{m-1}\rtimes \mathbb {Z}_2}\subset {{\tilde{K}}}\) is a maximal compact subgroup of \(\mathrm {Aut}({\mathcal {N}})\). We prove that, for odd \(m\ge 4\), there is no infra-nilmanifold which is essentially covered by the nilmanifold \(\Gamma _m\backslash {\mathcal {N}}_m\). For \(m=2n\ge 4\) (even), there is a unique infra-nilmanifold which is essentially (and doubly) covered by the nilmanifold \(\Gamma _m\backslash {\mathcal {N}}_m\).     

On kernels and nuclei of rank metric codes

For each rank metric code \(\mathcal {C}\subseteq \mathbb {K}^{m\times n}\), we associate a translation structure, the kernel of which is shown to be invariant with respect to the equivalence on rank metric codes. When \(\mathcal {C}\) is \(\mathbb {K}\)-linear, we also propose and investigate other two invariants called its middle nucleus and right nucleus. When \(\mathbb {K}\) is a finite field \(\mathbb {F}_q\) and \(\mathcal {C}\) is a maximum rank distance code with minimum distance \(d<\min \{m,n\}\) or \(\gcd (m,n)=1\), the kernel of the associated translation structure is proved to be \(\mathbb {F}_q\). Furthermore, we also show that the middle nucleus of a linear maximum rank distance code over \(\mathbb {F}_q\) must be a finite field; its right nucleus also has to be a finite field under the condition \(\max \{d,m-d+2\} \geqslant \left\lfloor \frac{n}{2} \right\rfloor +1\). Let \(\mathcal {D}\) be the DHO-set associated with a bilinear dimensional dual hyperoval over \(\mathbb {F}_2\). The set \(\mathcal {D}\) gives rise to a linear rank metric code, and we show that its kernel and right nucleus are isomorphic to \(\mathbb {F}_2\). Also, its middle nucleus must be a finite field containing \(\mathbb {F}_q\). Moreover, we also consider the kernel and the nuclei of \(\mathcal {D}^k\) where k is a Knuth operation.     

Malcev products of unipotent monoids and varieties of bands

Let \(\mathcal{U}\) be the class of all unipotent monoids and \(\mathcal{B}\) the variety of all bands. We characterize the Malcev product \(\mathcal{U} \circ \mathcal{V}\) where \(\mathcal{V}\) is a subvariety of \(\mathcal{B}\) low in its lattice of subvarieties, \(\mathcal{B}\) itself and the subquasivariety \(\mathcal{S} \circ \mathcal{RB}\), where \(\mathcal{S}\) stands for semilattices and \(\mathcal{RB}\) for rectangular bands, in several ways including by a set of axioms. For members of some of them we describe the structure as well. This succeeds by using the relation \(\widetilde{\mathcal{H}}= \widetilde{\mathcal{L}} \cap \widetilde{\mathcal{R}}\), where \(a\;\,\widetilde{\mathcal{L}}\;\,b\) if and only if a and b have the same idempotent right identities, and \(\widetilde{\mathcal{R}}\) is its dual.We also consider \((\mathcal{U} \circ \mathcal{RB}) \circ \mathcal{S}\) which provides the motivation for this study since \((\mathcal{G} \circ \mathcal{RB}) \circ \mathcal{S}\) coincides with completely regular semigroups, where \(\mathcal{G}\) is the variety of all groups. All this amounts to a generalization of the latter: \(\mathcal{U}\) instead of \(\mathcal{G}\).     

rthogonality Property $$\mathcal {}$$ and Compact Perturbations

A bounded linear operator T acting on a Hilbert space is said to have orthogonality property \(\mathcal {O}\) if the subspaces \(\ker (T-\alpha )\) and \(\ker (T-\beta )\) are orthogonal for all \(\alpha , \beta \in \sigma _p(T)\) with \(\alpha \ne \beta \). In this paper, the authors investigate the compact perturbations of operators with orthogonality property \(\mathcal {O}\). We give a sufficient and necessary condition to determine when an operator T has the following property: for each \(\varepsilon >0\), there exists \(K\in \mathcal {K(H)}\) with \(\Vert K\Vert <\varepsilon \) such that \(T+K\) has orthogonality property \(\mathcal {O}\). Also, we study the stability of orthogonality property \(\mathcal {O}\) under small compact perturbations and analytic functional calculus.     

A partition relation for pairs on $$\omega ^{\omega ^\omega }$$

We consider colorings of the pairs of a family \(\mathcal {F}\subseteq {{\mathrm{FIN}}}\) of topological type \(\omega ^{\omega ^k}\), for \(k>1\); and we find a homogeneous family \(\mathcal {G}\subseteq \mathcal {F}\) for each coloring. As a consequence, we complete our study of the partition relation \({\forall l>1,\, \alpha \rightarrow ({{\mathrm{top}}}\;\omega ^2+1)^2_{l,m}}\) identifying \(\omega ^{\omega ^\omega }\) as the smallest ordinal space \(\alpha <\omega _1\) satisfying \({\forall l>1,\, \alpha \rightarrow ({{\mathrm{top}}}\;\omega ^2+1)^2_{l,4}}\).     

On counting subring-submodules of free modules over finite commutative frobenius rings

Let \(\texttt {R}\) be a finite commutative Frobenius ring and \(\texttt {S}\) a Galois extension of \(\texttt {R}\) of degree m. For positive integers k and \(k'\), we determine the number of free \(\texttt {S}\)-submodules \(\mathcal {B}\) of \(\texttt {S}^\ell \) with the property \(k=\texttt {rank}_\texttt {S}(\mathcal {B})\) and \(k'=\texttt {rank}_\texttt {R}(\mathcal {B}\cap \texttt {R}^\ell )\). This corrects the wrong result (Bill in Linear Algebr Appl 22:223–233, 1978, Theorem 6) which was given in the language of codes over finite fields.     

Irreducible Morphisms and Locally Finite Dimensional Representations

Let \(\mathcal {A}\) be a Hom-finite additive Krull-Schmidt k-category where k is an algebraically closed field. Let \(\text {mod}\mathcal {A}\) denote the category of locally finite dimensional \(\mathcal {A}\)-modules, that is, the category of covariant functors \(\mathcal {A} \to \text {mod}k\). We prove that an irreducible monomorphism in \(\text {mod}\mathcal {A}\) has a finitely generated cokernel, and that an irreducible epimorphism in \(\text {mod}\mathcal {A}\) has a finitely co-generated kernel. Using this, we get that an almost split sequence in \(\text {mod}\mathcal {A}\) has to start with a finitely co-presented module and end with a finitely presented one. Finally, we apply our results to the study of rep(Q), the category of locally finite dimensional representations of a strongly locally finite quiver. We describe all possible shapes of the Auslander-Reiten quiver of rep(Q).     

Idempotent and <Emphasis Type="Italic">p</Emphasis>-potent quadratic functions: distribution of nonlinearity and co-dimension

The Walsh transform \(\widehat{Q}\) of a quadratic function \(Q:{\mathbb F}_{p^n}\rightarrow {\mathbb F}_p\) satisfies \(|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}\) for all \(b\in {\mathbb F}_{p^n}\), where \(0\le s\le n-1\) is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class \(\mathcal {C}_1\) is defined for arbitrary n as \(\mathcal {C}_1 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{\lfloor (n-1)/2\rfloor }a_ix^{2^i+1})\;:\; a_i \in {\mathbb F}_2\}\), and the larger class \(\mathcal {C}_2\) is defined for even n as \(\mathcal {C}_2 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{(n/2)-1}a_ix^{2^i+1}) + \mathrm{Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in {\mathbb F}_2\}\). For an odd prime p, the subclass \(\mathcal {D}\) of all p-ary quadratic functions is defined as \(\mathcal {D} = \{Q(x) = \mathrm{Tr_n}(\sum _{i=0}^{\lfloor n/2\rfloor }a_ix^{p^i+1})\;:\; a_i \in {\mathbb F}_p\}\). We determine the generating function for the distribution of the parameter s for \(\mathcal {C}_1, \mathcal {C}_2\) and \(\mathcal {D}\). As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case \(p > 2\), the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to \(\mathcal {C}_1\) and \(\mathcal {C}_2\) in terms of a generating function.     

On the image,characterization, and automatic continuity of $$\varvec{(\sigma }, \varvec{\tau }$$ )-derivations

The first main theorem of this paper asserts that any \((\sigma , \tau )\)-derivation d, under certain conditions, either is a \(\sigma \)-derivation or is a scalar multiple of (\(\sigma - \tau \)), i.e. \(d = \lambda (\sigma - \tau )\) for some \(\lambda \in \mathbb {C} \backslash \{0\}\). By using this characterization, we achieve a result concerning the automatic continuity of \((\sigma , \tau \))-derivations on Banach algebras which reads as follows. Let \(\mathcal {A}\) be a unital, commutative, semi-simple Banach algebra, and let \(\sigma , \tau : \mathcal {A} \rightarrow \mathcal {A}\) be two distinct endomorphisms such that \(\varphi \sigma (\mathbf e )\) and \(\varphi \tau (\mathbf e )\) are non-zero complex numbers for all \(\varphi \in \Phi _\mathcal {A}\). If \(d : \mathcal {A} \rightarrow \mathcal {A}\) is a \((\sigma , \tau )\)-derivation such that \(\varphi d\) is a non-zero linear functional for every \(\varphi \in \Phi _\mathcal {A}\), then d is automatically continuous. As another objective of this research, we prove that if \(\mathfrak {M}\) is a commutative von Neumann algebra and \(\sigma :\mathfrak {M} \rightarrow \mathfrak {M}\) is an endomorphism, then every Jordan \(\sigma \)-derivation \(d:\mathfrak {M} \rightarrow \mathfrak {M}\) is identically zero.     

Bavrin’s Type Factorization of the Temljakov Operator for Holomorphic Functions in Circular Domains of $$\mathbb {C}^{n}$$

The paper concerns investigations of holomorphic functions of several complex variables with a factorization of their Temljakov transform. Firstly, there were considered some inclusions between the families \(\mathcal {C}_{\mathcal {G}},\mathcal {M}_{\mathcal {G}},\mathcal {N}_{\mathcal {G}},\mathcal {R}_{\mathcal {G}},\mathcal {V}_{\mathcal {G}}\) of such holomorphic functions on complete n-circular domain \(\mathcal {G}\) of \(\mathbb {C}^{n}\) in some papers of Bavrin, Fukui, Higuchi, Michiwaki. A motivation of our investigations is a condensation of the mentioned inclusions by some new families of Bavrin’s type. Hence we consider some families \(\mathcal {K}_{ \mathcal {G}}^{k},k\ge 2,\) of holomorphic functions f :  \(\mathcal {G}\rightarrow \mathbb {C},f(0)=1,\) defined also by a factorization of \( \mathcal {L}f\) onto factors from \(\mathcal {C}_{\mathcal {G}}\) and \(\mathcal {M} _{\mathcal {G}}.\) We present some interesting properties and extremal problems on \(\mathcal {K}_{\mathcal {G}}^{k}\).     

Dynamics of Concentration in a Population Model Structured by Age and a Phenotypical Trait

In this paper, we consider the initial-boundary value problem of the two-species chemotaxis Keller-Segel model

$$\begin{aligned} \textstyle\begin{cases} u_{t}=\Delta u-\chi_{1}\nabla \cdot (u\nabla w), &x\in \varOmega , \ t>0, \\ v_{t}=\Delta v-\chi_{2}\nabla \cdot (v\nabla w), &x\in \varOmega , \ t>0, \\ 0=\Delta w-\gamma w+\alpha_{1}u+\alpha_{2}v, &x\in \varOmega , \ t>0, \end{cases}\displaystyle \end{aligned}$$

where the parameters \(\chi_{1}\), \(\chi_{2}\), \(\alpha_{1}\), \(\alpha_{2}\), \(\gamma \) are positive constants, \(\varOmega \subset \mathbb{R}^{2}\) is a bounded domain with smooth boundary. We obtain the results for finite time blow-up and global bounded as follows: (1) For any fixed \(x_{0}\in \varOmega \), if \(\chi_{1}\alpha_{2}= \chi_{2}\alpha_{1}\), \(\int_{\varOmega }(u_{0}+v_{0})|x-x_{0}|^{2}dx\) is sufficiently small, and \(\int_{\varOmega }(u_{0}+v_{0})dx>\frac{8\pi ( \chi_{1}\alpha_{1}+\chi_{2}\alpha_{2})}{\chi_{1}\alpha_{1}\chi_{2} \alpha_{2}}\), then the nonradial solution of the two-species Keller-Segel model blows up in finite time. Moreover, if \(\varOmega \) is a convex domain, we find a lower bound for the blow-up time; (2) If \(\|u_{0}\|_{L^{1}(\varOmega )}\) and \(\|v_{0}\|_{L^{1}( \varOmega )}\) lie below some thresholds, respectively, then the solution exists globally and remains bounded.

     

Quadratic programs with hollows

Let \(\mathcal {F}\) be a quadratically constrained, possibly nonconvex, bounded set, and let \(\mathcal {E}_1, \ldots , \mathcal {E}_l\) denote ellipsoids contained in \(\mathcal {F}\) with non-intersecting interiors. We prove that minimizing an arbitrary quadratic \(q(\cdot )\) over \(\mathcal {G}:= \mathcal {F}{\setminus } \cup _{k=1}^\ell {{\mathrm{int}}}(\mathcal {E}_k)\) is no more difficult than minimizing \(q(\cdot )\) over \(\mathcal {F}\) in the following sense: if a given semidefinite-programming (SDP) relaxation for \(\min \{ q(x) : x \in \mathcal {F}\}\) is tight, then the addition of l linear constraints derived from \(\mathcal {E}_1, \ldots , \mathcal {E}_l\) yields a tight SDP relaxation for \(\min \{ q(x) : x \in \mathcal {G}\}\). We also prove that the convex hull of \(\{ (x,xx^T) : x \in \mathcal {G}\}\) equals the intersection of the convex hull of \(\{ (x,xx^T) : x \in \mathcal {F}\}\) with the same l linear constraints. Inspired by these results, we resolve a related question in a seemingly unrelated area, mixed-integer nonconvex quadratic programming.     

Inflation of Finite Lattices Along All-or-nothing Sets

We introduce a new generalization of Alan Day’s doubling construction. For ordered sets \(\mathcal {L}\) and \(\mathcal {K}\) and a subset \(E \subseteq \ \leq _{\mathcal {L}}\) we define the ordered set \(\mathcal {L} \star _{E} \mathcal {K}\) arising from inflation of \(\mathcal {L}\) along E by \(\mathcal {K}\). Under the restriction that \(\mathcal {L}\) and \(\mathcal {K}\) are finite lattices, we find those subsets \(E \subseteq \ \leq _{\mathcal {L}}\) such that the ordered set \(\mathcal {L} \star _{E} \mathcal {K}\) is a lattice. Finite lattices that can be constructed in this way are classified in terms of their congruence lattices.A finite lattice is binary cut-through codable if and only if there exists a 0?1 spanning chain \(\left \{\theta _{i}\colon 0 \leq i \leq n \right \}\) in \(Con(\mathcal {L})\) such that the cardinality of the largest block of ?? _i /?? _i?1 is 2 for every i with 1≤i≤n. These are exactly the lattices that can be constructed by inflation from the 1-element lattice using only the 2-element lattice. We investigate the structure of binary cut-through codable lattices and describe an infinite class of lattices that generate binary cut-through codable varieties.     

A Characterization Theorem for the $$L^{2}$$-Discrepancy of Integer Points in Dilated Polygons

Let C be a convex d-dimensional body. If \(\rho \) is a large positive number, then the dilated body \(\rho C\) contains \(\rho ^{d}\left| C\right| +\mathcal {O}\left( \rho ^{d-1}\right) \) integer points, where \(\left| C\right| \) denotes the volume of C. The above error estimate \(\mathcal {O}\left( \rho ^{d-1}\right) \) can be improved in several cases. We are interested in the \(L^{2}\)-discrepancy \(D_{C}(\rho )\) of a copy of \(\rho C\) thrown at random in \(\mathbb {R}^{d}\). More precisely, we consider

Open image in new window

where \(\mathbb {T}^{d}=\) \(\mathbb {R}^{d}/\mathbb {Z}^{d}\) is the d-dimensional flat torus and \(SO\left( d\right) \) is the special orthogonal group of real orthogonal matrices of determinant 1. An argument of Kendall shows that \(D_{C}(\rho )\le c\ \rho ^{(d-1)/2}\). If C also satisfies the reverse inequality \(\ D_{C}(\rho )\ge c_{1} \ \rho ^{(d-1)/2}\), we say that C is \(L^{2}\) -regular. Parnovski and Sobolev proved that, if \(d>1\), a d-dimensional unit ball is \(L^{2} \)-regular if and only if \(d\not \equiv 1\ ({\text {mod}}4)\). In this paper we characterize the \(L^{2}\)-regular convex polygons. More precisely, we prove that a convex polygon is not \(L^{2}\)-regular if and only if it can be inscribed in a circle and it is symmetric about the centre.