The Ideal Structure of Semigroups of Linear Transformations with Upper Bounds on Their Nullity or Defect

Suppose V is a vector space with dim V = p ≥ q ≥ ?₀, and let T(V) denote the semigroup (under composition) of all linear transformations of V. For α ∈ T (V), let ker α and ran α denote the “kernel” and the “range” of α, and write n(α) = dim ker α and d(α) = codim ran α. In this article, we study the semigroups AM(p, q) = {α ∈ T(V):n(α) < q} and AE(p, q) = {α ∈ T(V):d(α) < q}. First, we determine whether they belong to the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. Then, for each semigroup, we describe its maximal regular subsemigroup, and we characterise its Green's relations and (two-sided) ideals. As a precursor to further work in this area,, we also determine all the maximal right simple subsemigroups of AM(p, q).     

Limit Theorems for Randomly Selected Ratios of Order Statistics from a Pareto Distribution

Abstract

Consider independent and identically distributed random variables {X _nk , 1 ≤ k ≤ m, n ≥ 1} from the Pareto distribution. We randomly select a pair of order statistics from each row, X _n(i) and X _n(j), where 1 ≤ i < j ≤ m. Then we test to see whether or not Strong and Weak Laws of Large Numbers with nonzero limits for weighted sums of the random variables X _n(j)/X _n(i) exist where we place a prior distribution on the selection of each of these possible pairs of order statistics.     

0-Minimal Quasi-ideals of Generalized Linear Transformation Semigroups

Abstract

For vector spaces V and W over a field F, L _F (V, W) denotes the set of all linear transformations α : V → W, and for a cardinal number k > 0, let L _F (V, W, k) be the set of all α ∈ L _F (V, W) of rank less than k. For θ ∈ L _F (W, V), let (L _F (V, W, k), θ) denote the semigroup L _F (V, W, k) under the operation ? defined by α ? β = αθβ for all α, β ∈ L _F (V, W, k). In this paper, all 0-minimal quasi-ideals of the semigroup (L _F (V, W, k), θ) are completely characterized. It is also shown from this characterization that every nonzero semigroup (L _F (V, W, k), θ) always has a 0-minimal quasi-ideal.     

Maximal and minimal congruences on some semigroups

In 2006, Sanwong and Sullivan described the maximal congruences on the semigroup N consisting of all non-negative integers under standard multiplication, and on the semigroup T（X） consisting of all total transformations of an infinite set X under composition. Here, we determine all maximal congruences on the semigroup Zn under multiplication modulo n. And, when Y lohtain in X, we do the same for the semigroup T（X, Y） consisting of all elements of T（X） whose range is contained in Y. We also characterise the minimal congruences on T（X. Y）.     

Prophet Regions for Independent [0, 1]-Valued Random Variables with Random Discounting

Abstract

Let X ₁, X ₂… and B ₁, B ₂… be mutually independent [0, 1]-valued random variables, with EB _j  = β > 0 for all j. Let Y _j  = B ₁ … sB _j?1 X _j for j ≥ 1. A complete comparison is made between the optimal stopping value V(Y ₁,…,Y _n ):=sup{EY _τ:τ is a stopping rule for Y ₁,…,Y _n } and E(max _1≤j≤n Y _j ). It is shown that the set of ordered pairs {(x, y):x = V(Y ₁,…,Y _n ), y = E(max _1≤j≤n Y _j ) for some sequence Y ₁,…,Y _n obtained as described} is precisely the set {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ Ψ _{n, β}(x)}, where Ψ _{n, β}(x) = [(1 ? β)n + 2β]x ? β^?(n?2) x ² if x ≤ β ^n?1, and Ψ _{n, β}(x) = min _j≥1{(1 ? β)jx + β ^j } otherwise. Sharp difference and ratio prophet inequalities are derived from this result, and an analogous comparison for infinite sequences is obtained.     

On Finite n-Acentralizer Groups

Let G be a group and Aut(G) be the group of automorphisms of G. Then the Acentralizer of an automorphism α ∈Aut(G) in G is defined as C _G (α) = {g ∈ G∣α(g) = g}. For a finite group G, let Acent(G) = {C _G (α)∣α ∈Aut(G)}. Then for any natural number n, we say that G is n-Acentralizer group if |Acent(G)| =n. We show that for any natural number n, there exists a finite n-Acentralizer group and determine the structure of finite n-Acentralizer groups for n ≤ 5.     

Abundance of Semigroups of Transformations Restricted by an Equivalence

Let σ be an equivalence on X, and let E(X, σ) denote the semigroup (under composition) of all f: X → X such that σ ? ker(f). In this article, we show that the semigroup E(X, σ) is right abundant but not left abundant whenever |X| ≥3 and σ is non-trivial.     

Generating Sets of Finite Transformation Semigroups PK(n,r) and K (n,r)

Let P _n and T _n be the partial transformation and the full transformation semigroups on the set {1,…, n}, respectively. In this paper we find necessary and sufficient conditions for any set of partial transformations of height r in the subsemigroup PK(n, r) = {α ∈P _n : |im (α)| ≤r} of P _n to be a (minimal) generating set of PK(n, r); and similarly, for any set of full transformations of height r in the subsemigroup K(n, r) = {α ∈T _n : |im (α)| ≤r} of T _n to be a (minimal) generating set of K(n, r) for 2 ≤ r ≤ n ? 1.     

On Smash Products Of Hopf Algebras

Let H = X? _R A denote an R-smash product of the two bialgebras X and A. We prove that (X,A) is a pair of matched bialgebras, if the R-smash product H has a braiding structure. When X is an associative algebra and A is a Hopf algebra, we investigate the global dimension and the weak dimension of the smash product H = X? _R A and show that lD(H) ≤ rD(A) + lD(X) and wD(H) ≤ wD(A) + wD(X). As an application, we get lD(H ₄) = ∞ for Sweedler's four dimensional Hopf algebra H ₄. We also study the associativity of smash products and the relations between smash products and factorization for algebras.     

Ample Vector Bundles with Small g − q

Let X be a smooth complex projective variety and let Z ? X be a smooth submanifold of dimension ≥ 2, which is the zero locus of a section of an ample vector bundle ? of rank dim X ? dim Z ≥ 2 on X. Let H be an ample line bundle on X, whose restriction H _Z to Z is generated by global sections. The structure of triplets (X,?,H) as above is described under the assumption that the curve genus of the corank-1 vector bundle ? ⊕ H ^{⊕ (dim Z?1)} is ≤ h ¹( _X ) + 2.     

On B(n,k) 2-groups

A finite group G is said to be a B(n, k) group if for any n-element subset {a ₁,…, a _n } of G, |{a _i a _j |1 ≤ i, j ≤ n}| ≤k. In this article, we give characterizations of the B(5, 19) 2-groups, and the B(6, k) 2-groups for 21 ≤ k ≤ 28.     

Idempotent Generation in the Endomorphism Monoid of a Uniform Partition

Denote by 𝒯_n and 𝒮_n the full transformation semigroup and the symmetric group on the set {1,…, n}, and ?_n = {1} ∪ (𝒯_n?𝒮_n). Let 𝒯(X, 𝒫) denote the monoid of all transformations of the finite set X preserving a uniform partition 𝒫 of X into m subsets of size n, where m, n ≥ 2. We enumerate the idempotents of 𝒯(X, 𝒫), and describe the submonoid S = ? E ? generated by the idempotents E = E(𝒯(X, 𝒫)). We show that S = S₁ ∪ S₂, where S₁ is a direct product of m copies of ?_n, and S₂ is a wreath product of 𝒯_n with 𝒯_m?𝒮_m. We calculate the rank and idempotent rank of S, showing that these are equal, and we also classify and enumerate all the idempotent generating sets of minimal size. In doing so, we also obtain new results about arbitrary idempotent generating sets of ?_n.     

Reductivity of the Lie Algebra of a Bilinear Form

Let f: V × V → F be a totally arbitrary bilinear form defined on a finite dimensional vector space V over a field F, and let L(f) be the subalgebra of 𝔤𝔩(V) of all skew-adjoint endomorphisms relative to f. Provided F is algebraically closed of characteristic not 2, we determine all f, up to equivalence, such that L(f) is reductive. As a consequence, we find, over an arbitrary field, necessary and sufficient conditions for L(f) to be simple, semisimple or isomorphic to 𝔰𝔩(n) for some n.     

A Generalization of a Lemma of Sullivan

Consider an irreducible polynomial of the form f(X) = X ^p  ? aX ? b ∈ 𝔽[X] and α a root of f(X), where 𝔽 is a field of characteristic p. In 1975, F.J. Sullivan stated a lemma that provides the trace, taken with respect to the extension 𝔽(α)/𝔽, of elements of the form α ⁿ , where 0 ≤ n ≤ p ² ? 1. We present a generalization of Sullivan's Lemma and provide another proof of the original lemma. We explain how computing Tr(α ⁿ ) for n < p ^r can be reduced to computing the traces Tr(α ^m ) for all m ≤ r(p ? 1).     

On Polarized 3-Folds (X,L) Such That <lowercase > h < /lowercase >0(L) = 2 and the Sectional Genus of (X,L) is Equal to the Irregularity of X

Let X be a smooth complex projective variety of dimension 3 and let L be an ample line bundle on X. In this article, we give a characterization of (X, L) with g(X, L) = q(X) and h⁰(L) = 2, where g(X, L) (resp. q(X)) denotes the sectional genus of (X, L) (resp. the irregularity of X).     

Remarks on Taylor Series Expansions and Conditional Expectations for Stratonovich SDEs with Complete V‐Commutativity

Abstract

It may happen that there is not a finite maximum order bound for numerical approximations of stochastic processes X = (X _t : 0 ≤ t ≤ T) satisfying Stratonovich stochastic differential equations (SDEs) with some commutative structure along an appropriate functional V(t, X _t ). This statement can be proven with respect to the concept of mean square convergence under the assumption of “infinite smoothness” of drift a(t, x) and diffusion coefficients b ^j (t, x) and with finite initial second moments. As a result, we obtain an infinite series expansion of the conditional expectation 𝔼[V(t, X _t )|? _t ^N ] on any fixed finite time interval [0, T], provided that the information is collected by discretized σ‐field ? _T ^N  = σ{W _{t 0} , W _{t 1} , …, W _{t N?1} , W _T } at N + 1 given time instants t _i  ∈ [0, T] with t ₀ ≤ t ₁ ≤ ··· ≤ t _N?1 ≤ t _N  = T.     

Semistability of Logarithmic Cotangent Bundle on Some Projective Manifolds

In this article, we investigate the semistability of logarithmic de Rham sheaves on a smooth projective variety (X, D), under suitable conditions. This is related to existence of Kähler–Einstein metric on the open variety. We investigate this problem when the Picard number is one. Fix a normal crossing divisor D on X and consider the logarithmic de Rham sheaf Ω _X (log D) on X. We prove semistability of this sheaf, when the log canonical sheaf K _X  + D is ample or trivial, or when ?K _X  ? D is ample, i.e., when X is a log Fano n-fold of dimension n ≤ 6. We also extend the semistability result for Kawamata coverings, and this gives examples whose Picard number can be greater than one.     

The multiplier ideal is a universal test ideal

Abstract

For a canonical threefold X, we know that h ⁰(X, 𝒪 _X (nK _X )) ≥ 1 for a sufficiently large n. When χ(𝒪 _X ) > 0, it is not easy to get such an integer n. Fletcher showed that h ⁰(X, 𝒪 _X (12K _X )) ≥ 1 and h ⁰(X, 𝒪 _X (24K _X )) ≥ 2 when χ(𝒪 _X ) = 1. He inquired about existence of a canonical threefold with given conditions which shows the result sharp. We show that such an example does not exist. Using a different technique, we prove h ⁰(X, 𝒪 _X (12K _X )) ≥ 2.     

A Note on Semigroup Algebras of Permutable Semigroups

Let S be a semigroup and 𝔽 be a field. For an ideal J of the semigroup algebra 𝔽[S] of S over 𝔽, let ?_J denote the restriction (to S) of the congruence on 𝔽[S] defined by the ideal J. A semigroup S is called a permutable semigroup if α ○ β = β ○ α is satisfied for all congruences α and β of S. In this paper we show that if S is a semilattice or a rectangular band then φ_{{S; 𝔽}}: J → ?_J is a homomorphism of the semigroup (Con(𝔽[S]); ○ ) into the relation semigroup (?_S; ○ ) if and only if S is a permutable semigroup.