Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in 𝕊3. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using v-move and forbidden moves. In this paper, we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in

A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field 𝔽 can be reduced to the following three classifications, for

For n,m∈ℕ, let Hn,m be the dual of the Radford algebra of dimension n2m. We present new finite-dimensional Nichols algebras arising from the study of simple Yetter–Drinfeld modules over Hn,m. Along the way, we describe the simple objects in Hn,mHn,m𝒴𝒟 and their projective envelopes. Then we determine those simple modules that give rise to finite-dimensional Nichols algebras for the case n=2. There

We investigate the representations of the hyperalgebras associated to the map algebras 𝔤⊗𝒜, where 𝔤 is any finite-dimensional complex simple Lie algebra and 𝒜 is any associative commutative unitary algebra with a multiplicatively closed basis. We consider the natural definition of the local and global Weyl modules, and the Weyl functor for these algebras. Under certain conditions, we prove that

We compute all liftings of braidings of unidentified types 𝔲𝔣𝔬(7) and 𝔲𝔣𝔬(8). Some of these examples had been previously computed by Helbig. We also present a list of examples of liftings of modular type 𝔟𝔯(2,a). Watching a coast as it slips by the ship is like thinking about an enigma. There it is before you, smiling, frowning, inviting, grand, mean, insipid, or savage, and always mute with

Let Fq be a finite field with q elements, n≥2 a positive integer, T(n,q) the semigroup of all n×n upper triangular matrices over Fq under matrix multiplication, T∗(n,q) the group of all invertible matrices in T(n,q), PT∗(n,q) the quotient group of T∗(n,q) by its center. The one-divisor graph of T(n,q), written as 𝕋n(q), is defined to be a directed graph with T(n,q) as vertex set, and there is a directed

Let A be a finite-dimensional algebra. If A is self-injective, then all modules are reflexive. Marczinzik recently has asked whether A has to be self-injective in case all the simple modules are reflexive. Here, we exhibit an 8-dimensional algebra which is not self-injective, but such that all simple modules are reflexive (actually, for this example, the simple modules are the only non-projective indecomposable

We give a geometric classification of complex 4-dimensional nilpotent ℭ𝔇-algebras. The corresponding geometric variety has dimension 18 and decomposes into 2 irreducible components determined by the Zariski closures of a two-parameter family of algebras and a four-parameter family of algebras. In particular, there are no rigid 4-dimensional complex nilpotent ℭ𝔇-algebras.

In this paper, we introduce the concept of Baer (p,q)-sets. Using this notion, we define Rickart, Baer, quasi-Baer and π-Baer (S,R)-bimodules, respectively. We show how these conditions relate to each other. We also develop new properties of the minus binary relation, ≤-, we extend the relation ≤- to (S,R)-bimodules and use it to characterize the aforementioned Rickart, Baer, quasi-Baer, and π-Baer

In this paper, we calculate the combinatorial rank of the positive part uq+(𝔤) of the multiparameter version of the small Lusztig quantum group, where 𝔤 is a simple Lie algebra of type G2. Supposing that the main parameter of quantization q has multiplicative order t, where t is finite, t>3, we prove that the combinatorial rank equals 3.

Let 𝔽 denote a field with char 𝔽≠2. The Racah algebra ℜ is the unital associative 𝔽-algebra defined by generators and relations in the following way. The generators are A, B, C, D. The relations assert that [A,B]=[B,C]=[C,A]=2D and each of the elements α=[A,D]+AC−BA,β=[B,D]+BA−CB,γ=[C,D]+CB−AC is central in ℜ. Additionally, the element δ=A+B+C is central in ℜ. We call each element in D2+A2+B2+(δ+2){A

Let R be a polynomial ring in n indeterminates with coefficients in the field K of characteristic p>0 and 𝒟 be the ring of differential operators over R. In this paper, we introduce the notion of generalized Eulerian 𝒟-modules for characteristic p>0 and establish their properties. We show that if 𝒯 is any graded Lyubeznik functor on the category of modules over R, then 𝒯(R) is a generalized Eulerian

The Manturov (3,4)-group G43 is the group generated by four elements a, b, c, d with defining relations a2=b2=c2=d2=(abcd)2=(acdb)2=(adbc)2=1. We apply the algebraic discrete Morse theory to calculate the Anick chain complex for G43, evaluate the Hochschild cohomology groups of the group algebra 𝕂G43 with coefficients in all 1-dimensional bimodules over a field 𝕂 of characteristic zero, and derive

A Kuribayashi quartic curve 𝒞a:X4+Y4+Z4+a(X2Y2+Y2Z2+Z2X2)=0,a∈ℂ∖{−1,±2}, carries total sextactic points if and only if a=14 or a is a zero of P(a)=a3+68a2−91a+98, cf. [1]. In [2], the authors describe the subgroup generated by the total sextactic points in the Jacobian of a Kuribayashi quartic curve when a is a zero of P(a). In this paper, we describe this group when a=14.

The aim of this paper is to introduce and study Yetter–Drinfeld category over a weak monoidal Hom–Hopf algebra H. We first show that the category HH𝒲ℳℋ𝒴𝒟 of Yetter–Drinfeld modules over H with a bijective antipode is a braided monoidal category. Secondly, we discuss some properties on the symmetries of the category HH𝒲ℳℋ𝒴𝒟. Finally, we prove that the representation category of triangular weak

A good way of parameterizing zero-dimensional schemes in an affine space 𝔸Kn has been developed in the last 20 years using border basis schemes. Given a multiplicity μ, they provide an open covering of the Hilbert scheme Hilbμ(𝔸Kn) and can be described by easily computable quadratic equations. A natural question arises on how to determine loci which are contained in border basis schemes and whose

In this paper, we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell (C∗-algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be regarded as the sectional algebras of trivial (direct product) bundles. Several theorems which relate geometric and algebraic constructions — via the construction of

We find sufficient conditions guaranteeing that for a quasivariety M of structures of finite type containing a B-class with respect to M, there exists a subquasivariety K⊆M and a structure 𝒜∈K such that the problems whether a finite lattice embeds into the lattice Lv(K) of K-varieties and into the lattice ConK𝒜 are undecidable.

The main result of this paper is the following: for all b ∈ ℤ there exists k = k(b) such that max{|A(k)|,|(A + u)(k)|}≥|A|b, for any finite A ⊂ ℚ and any nonzero u ∈ ℚ. Here, |A(k)| denotes the k-fold product set {a1⋯ak : a1,…,ak ∈ A}. Furthermore, our method of proof also gives the following l∞ sum-product estimate. For all γ > 0 there exists a constant C = C(γ) such that for any A ⊂ ℚ with |AA|≤

We study Heath-Brown’s and Serre’s dimension growth conjecture (proved by Salberger) when the degree d grows. Recall that Salberger’s dimension growth results give bounds of the form OX,𝜀(Bdim X+𝜀) for the number of rational points of height at most B on any integral subvariety X of ℙℚn of degree d ≥ 2, where one can write Od,n,𝜀 instead of OX,𝜀 as soon as d ≥ 4. We give the following simplified

We investigate the finitary functions from a finite field \(\mathbb {F}_q\) to the finite field \(\mathbb {F}_p\), where p and q are powers of different primes. An \((\mathbb {F}_p,\mathbb {F}_q)\)-linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through

Let H be a complex Hilbert space, and A be a positive bounded linear operator on H . Let B A ( H ) denote the set of all bounded linear operators on H whose A-adjoint exists. Let A denote a 2 × 2 operator matrix of the form A O O A . Very recently, for a strictly positive operator A, Bhunia et al. [On inequalities for A-numerical radius of operators. Electron J Linear Algebra. 2020;36:143–157] proved

The support poset of a monomial ideal \(I\subseteq {{\mathbf {k}}}[x_1,\ldots ,x_n]\) encodes the relation between the variables \(x_1,\ldots ,x_n\) and the minimal monomial generators of I. It is known that not every poset is realizable as the support poset of some monomial ideal. We describe some posets P for which we can explicitly find at least one monomial ideal \(I_P\) such that P is the support

In this paper, we consider a version of the fractional Clifford–Fourier transform (FrCFT) and study its several properties and applications to partial differential equations in Clifford analysis. First, we give the definition of the FrCFT and its inverse transform in the form of integral. Then, we discuss the relationship between the FrCFT and the Clifford–Fourier transform (CFT) and give some properties

Let G be an undirected graph with adjacency matrix A ( G ) . For a vertex u ∈ V ( G ) , the subgraph centrality of u in G is C S ( u ) = ( exp ⁡ ( A ( G ) ) ) u u = ∑ k = 0 ∞ 1 k ! ( A ( G ) k ) u u . In this paper, we give some formulas for C S ( u ) by using equitable partitions and star sets of G.

Given an ST-triple \((\mathcal {C},\mathcal {D},M)\) one can associate a co-t-structure on \(\mathcal {C}\) and a t-structure on \(\mathcal {D}\). It is shown that the discreteness of \(\mathcal {C}\) with respect to the co-t-structure is equivalent to the discreteness of \(\mathcal {D}\) with respect to the t-structure. As a special case, the discreteness of \(\mathcal {D}^{b}(\textsf {mod} A)\) in

In this paper, we first study the Gorenstein projective/flat dimension of complexes of modules. The relation between the Gorenstein projective/flat dimension for complexes and that for modules are investigated. Then we study Tate, stable and unbounded homology for complexes of modules. In the case of module arguments, we get some results that improve the known results.

We provide an explicit algorithm to compute a lifted Massey triple product relative to a defining system for a smooth projective plane curve X defined by a homogeneous polynomial G(x̲) over a field. The main idea is to use the description (due to Carlson and Griffiths) of the cup product for H1(X,ℂ) in terms of the multiplications inside the Jacobian ring of G(x̲) and the Cech–deRham complex of X.

We find a method for constructing DNA codes with single-deletion-correcting capability. We first present an explicit algorithm for the construction of the q-ary single-deletion-correcting codes (abbreviated as SDC codes) using a class of the complementary information set codes (abbreviated as CIS codes), where q is a power of a prime. We then show that the encoding/decoding scheme of the CIS codes

In this paper, we introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function \(e^x\) from classical differential calculus. A differential exponential map is an endomorphism which is compatible with the differential combinator in such a way that generalizations of \(e^0 = 1\), \(e^{x+y} = e^x e^y\), and \(\frac{\partial e^x}{\partial x} =

A power series is called lacunary if “almost all” of its coefficients are zero. Integer partitions have motivated the classification of lacunary specializations of Han’s extension of the Nekrasov–Okounkov formula. More precisely, we consider the modular forms $$\begin{aligned}F_{a,b,c}(z) :=\frac{\eta (24az)^a \eta (24acz)^{b-a}}{\eta (24z)},\end{aligned}$$ defined in terms of the Dedekind \(\eta \)-function

This paper presents an axisymmetric boundary element analysis of one-layered transversely isotropic material of halfspace extent with either ellipsoidal or spherical cavity. This analysis uses the fundamental solution of a transversely isotropic bi-material fullspace subject to the body force uniformly concentrated at a circular ring. The finite and infinite boundary elements are, respectively, used

In this paper, extending the idea presented by Takeuchi in [M. Takeuchi, Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra9 (1981) 841–882.] and more generally by Majid in [S. Majid, Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebra130(1) (1990) 17–64.], we introduce the notion of partial matched pair

Given a representation of a 3-Lie algebra, we construct a Lie 3-algebra, whose Maurer-Cartan elements are relative Rota-Baxter operators on the 3-Lie algebra. We define the cohomology of relative Rota-Baxter operators on 3-Lie algebras, by which we study deformations of relative Rota-Baxter operators. We show that if two formal deformations of a relative Rota-Baxter operator on a 3-Lie algebra are

Suppose that Z is a locally compact Hausdorff space and Ψ , Φ : E ⟶ F are C 0 ( Z ) -module maps between Hilbert C 0 ( Z ) -modules such that for every x , y ∈ E , x ⊥ y implies Ψ ( x ) ⊥ Φ ( y ) . Then there exists a bounded complex-valued function φ on Z that is continuous on Z E = { z ∈ Z : ⟨ x , x ⟩ ( z ) ≠ 0   f o r s o m e   x ∈ E } and satisfies ⟨ Ψ ( x ) , Φ ( y ) ⟩ = φ ⋅ ⟨ x , y ⟩ o n Z ,

In this paper, we study a fast algorithm for the numerical solution of the 1D distributed-order space-fractional diffusion equation. After discretization by the finite difference method, the resulting system is the symmetric positive definite Toeplitz matrix. The preconditioned conjugate gradient method with a circulant preconditioner is employed to solve the linear system. Theoretically, the spectrum

The present paper provides a procedure for constructing full-spark Parseval frames arising from the linear action of a 1-parameter group acting in R n . Precisely, given a square matrix A of order n, with real entries, we say that A induces the full spark frame property if the following conditions hold. There exists a vector v for which given any finite set X ⊂ R of cardinality n, the collection {

In this paper, we first define a two-sided Clifford Fourier transform(CFT) and its inverse transformation on \(L^{1}\) space. Then we study the differential of the two-sided CFT, the k-th power of \(F \{h\}\), Plancherel identity and time-frequency shift of the two-sided CFT. Finally we discuss the uncertainty principle of the two-sided CFT and give an application of the two-sided CFT to a partial

This study improved two parts of the virtual element method (VEM) to ensure that the stability of a stony soil slope, especially under excavation conditions, could be better studied. One of the improvements was the specific realization of the excavation algorithm, including the determination of how to identify polygonal elements that needed to be excavated, and the “deactivate and reactivate polygonal

It is known that there exists a surjective map from the set of weak (1, 3) homotopy classes of knot projections to the set of positive knots [N. Ito and Y. Takimura, (1, 2) and weak (1, 3) homotopies on knot projections, J. Knot Theory Ramifications22 (2013) 1350085]. An interesting question whether this map is also injective, which question was formulated independently by S. Kamada and Y. Nakanishi

A finitely generated group G acting on a tree with infinite cyclic edge and vertex stabilizers is called a generalized Baumslag–Solitar group (GBS group). We prove that a one-knot group G is a GBS group if and only if G is a torus knot group, and describe all n-knot GBS groups for n≥3.

In [V. O. Manturov, An almost classification of free knots, Dokl. Math.88(2) (2013) 556–558.] the second author constructed an invariant which in some sense generalizes the quantum sl(3) link invariant of Kuperberg to the case of free links. In this paper, we generalize this construction to free graph-links. As a result, we obtain an invariant of free graph-links with values in linear combinations

In the 1950s Milnor defined a family of higher-order invariants generalizing the linking number. Even the first of these new invariants, the triple linking number, has received fruitful study since its inception. In the case that a link L has vanishing pairwise linking numbers, this triple linking number gives an integer-valued invariant. When the linking numbers fail to vanish, the triple linking

We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by terms of the lower central series of knot groups. Also, we study decomposition of virtual

In 2003, Ozsváth and Szabó defined the concordance invariant τ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of τ for knots in S3 and a combinatorial proof that τ gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory

Ng constructed an invariant of knots in ℝ3, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ℝ4 using marked graph diagrams.

A twisted torus knot is a torus knot with some consecutive strands twisted. More precisely, a twisted torus knot T(p,q,r,s) is a torus knot T(p,q) with r consecutive strands s times fully twisted. We determine which twisted torus knots T(p,q,p−kq,−1) are a torus knot.

For any classical knot k1, we can construct a ribbon 2-knot spun(k1) by spinning an arc removed a small segment from k1 about R2 in R4. A ribbon 2-knot is an embedded 2-sphere in R4. If k1 has an n-crossing presentation, by spinning this, we can naturally construct a ribbon presentation with n ribbon crossings for spun(k1). Thus, we can define naturally a notion on ribbon 2-knots corresponding to the

Suppose that G is a finite group and H is a subgroup of G. H is said to be an SS-quasinormal subgroup of G if there is a subgroup B of G such that G=HB and H permutes with every Sylow subgroup of B. In this note, we fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1<|D|<|P| and study the p-nilpotency of G under the assumption that every subgroup H of P with |H|=|D| is SS-quasinormal

We introduce and study (strongly) π-Rickart objects and their duals in abelian categories, which generalize (strongly) self-Rickart objects and their duals. We establish general properties of such objects, we analyze their behavior with respect to coproducts, and we study classes all of whose objects are (strongly) π-Rickart. We derive consequences for module and comodule categories.

We study the maximum Hamming distance (or rather, the complementary notion of “minimum approximability”) of a general function on a finite group G to either of the sets End(G) and Aff(G), of group endomorphisms of G and affine maps on G, respectively, the latter being a certain generalization of endomorphisms. We give general bounds on these two quantities and discuss an infinite class of extremal

A finite p-group S is said to be of supersoluble type if every fusion system over S is supersoluble. The main aim of this paper is to characterise the finite p-groups of supersoluble type. Abelian and metacyclic p-groups of supersoluble type are completely described. Furthermore, we show that the Sylow p-subgroups of supersoluble type of a finite simple group must be cyclic.

In this paper we discuss finite presentability of the universal central extensions of Lie algebras sln(R), where n≥3 and R is a unital associative k-algebra. We show that a universal central extension is finitely presented if and only if the algebra R is finitely presented.

Let g be a hyperbolic Kac-Moody algebra of rank 2. We give a necessary and sufficient condition for the crystal graph of the Lakshmibai-Seshadri path crystal B ( λ ) to be connected for an arbitrary integral weight λ.

We improve a result of Isaacs et al. with respect to character degrees and class sizes of finite groups [1, Corollary D].

In this paper, a meshless BEM based on the radial integration method is developed to solve transient non-homogeneous convection-diffusion problem with spatially variable velocity and time-dependent source term. The Green function served as the fundamental solution is adopted to derive the boundary domain integral equation about the normalized field quantity. The two-point backward finite difference

We extend results of Positive Twist Knots and Thickenings [W. George, Positive twist knots and thickenings, J. Knot Theory Ramifications22 (2013) 1350046] to show that positive twist knots 𝒦m, for m≥3, satisfy the Uniform Thickness Property.

The aim of this paper is twofold. First, to provide a characterization of clopen topologies. We show that a topology is clopen if and only if it is symmetric and Alexandroff. Second, to investigate when some special unitary subrings of a power set define a (clopen) topology.

In this paper, by taking the class of all C3 (or D3) right R-modules for general envelopes and covers, we characterize a semisimple artinian ring (or a right perfect ring) via D3-covers (or D3-envelopes) and a right V-ring (or a right noetherian V-ring) via C3-covers (or C3-envelopes). By using isosimple-projective preenvelope, we obtained that if R is a semiperfect right FGF ring (or left coherent

Let \({\mathbb {F}}_{q}\) be a finite field of odd characteristic. We study Rédei functions that induce permutations over \(\mathbb {P}^1({\mathbb {F}}_{q})\) whose cycle decomposition contains only cycles of length 1 and j, for an integer \(j\ge 2\). When j is 4 or a prime number, we give necessary and sufficient conditions for a Rédei permutation of this type to exist over \(\mathbb {P}^1({\mathbb