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A Fredholm representation on a Hilbert space, whose kernel coincides with the ideal of compact operators, is constructed for the C∗-algebra B generated by all multiplication operators by piecewise ...

Let D be a digraph and and two subsets of where = {P: P is a non trivial finite path in D}. A subset N of V(D) is said to be an ()-kernel of D if: (1) for every {u,v} N there exists no uv-path P such that P (N is -independent), (2) for every vertex x in V(D) there exist y in N and P in such that P is an xy-path (N is -absorbent). As a particular case, the concept of ()-kernel generalizes the concept of kernel when = = A(D). A classical result in kernel theory is Richardson’s theorem which establishes that if D is a finite digraph without odd cycles, then D has a kernel. In this paper, the original results are sufficient conditions for the existence of ()-kernels in possibly infinite digraphs, in particular we will present some generalizations of Richardson’s theorem for infinite digraphs. Also we will deduce some conditions for the existence of kernels by monochromatic paths, H-kernels and (k,l)-kernels in possibly infinite digraphs.

In this paper, we define a transform which has the kernel in its definition and a concept of derivative for functionals on Wiener space. We then establish some results and formulas for the transfor...

We construct a basis of the kernel of the Lie bracketing map restricted to the submodule of the free associative algebra generated by all words having exactly one occurrence of both of two fixed le...

This work gives complete expressions of inertia of matrices involved in a linear subspace of R n × R n when they are presented in two different ways, specifically, as image and as well as kernel of...

Generalized Abel type integral equations with Gauss, Kummer's and Humbert's confluent hypergeometric functions in the kernel and generalized Abel type integral equations with localized fractional integrals are considered. The left-hand sides of these equations are inversed by using generalized fractional derivatives. Explicit solutions of the equations in the class of locally summable functions are obtained. They are represented in terms of hypergeometric functions. Asymptotic power exponential type expansions of the generalized and localized fractional integrals are obtained. The base solutions of the generalized Abel type integral equation are given in the form of asymptotic series.

The tangential k-Cauchy–Fueter operator and the k-CF functions on the quaternionic Heisenberg group are quaternionic counterparts of the tangential CR operator and CR functions on the Heisenberg group in the theory of several complex valuables. We use the group Fourier transform on the quaternionic Heisenberg group to analyze the operator associated the tangential k-Cauchy–Fueter operator and to construct its kernel, from which we get the Szego kernel of the orthogonal projection from the space of functions to the space of integrable k-CF functions on the quaternionic Heisenberg group.

In this paper, we suppose that and are two rings, where has a unit element 1 and , we show that if every Jordan homomorphism from into is a homomorphism (anti-homomorphism), then every n-Jordan homomorphism from into is an n-homomorphism (n-anti-homomorphism).

In this paper, we give the definition of T-operator with bihypermonogenic kernel in Clifford analysis and discuss a series of properties of this operator, such as uniform boundness, Holder continuity and - integrability. T-operator is a singular integral operator which is defined in the n-dimensional Euclidean space valued in the noncommutative Clifford algebra. The properties of T-operator play an important role in solving differential equations.

We investigate several properties of Aluthge transform of an operator . We prove (i) if T is (s, p)-w-hyponormal operator and is quasinormal (resp., normal), then T is quasinormal (resp., normal), (ii) if T is (s, p)-w-hyponormal operator and is a partial isometry, then T is quasinormal partial isometry, (iii) if T and are (s, p)-w-hyponormal operator, then T is normal, and (iv) Fuglede–Putnam type theorem holds for a class p-w-hyponormal operator T with if T satisfies a kernel condition

Let A be a finite group of odd order and let A act on a finite p-group P with |P|>pe, where e is an integer e≥4(e≥5 if p = 2). In this paper we show that P is centralized by Op(A𝔄p2−1) if every non-meta-cyclic subgroup of order pe in P is stabilized by Op(A). As applications, some conditions are given for a finite group G with the p-length ≤1 and the p-rank ≤2. We also find a class of finite p-groups, which is not only very useful for the paper but also has its independent meaning.

In this paper are defined cohomology-like groups that classify loop extensions satisfying a given identity in three variables for association identities and in two variables for the case of commutativity. It is considered a large amount of identities. These groups generalize those defined in Nishigori [3] and of Kenneth and Leedham-Green [2]. It is computed the number of metacyclic extensions for trivial action of the quotient on the kernel in one particular case for left Bol loops and in general for commutative loops.

This paper investigates affine control systems with jumps for which the ideal If(g1, …, gm) generated by the drift vector field f in the Lie algebra L(f, g1, …, gm) can be imbedded as a kernel of a linear first-order partial differential equation. It will lead us to uncontrollable affine control systems with jumps for which the corresponding reachable sets are included in explicitly described differentiable manifolds.

Let G be a p-adic Lie group with Lie algebra 𝔤 and Ad: G → Aut(𝔤) be the adjoint representation. It was claimed in the literature that the kernel K≔ker(Ad) always has an abelian open normal subgroup. We show by means of a counterexample that this assertion is false. It can even happen that K = G, but G has no abelian subnormal subgroup except for the trivial group. The arguments are based on auxiliary results on subgroups of free products with central amalgamation.

We compute the leading and subleading terms in the asymptotic expansion of the Szego kernel on the diagonal of a class of pseudoconvex Reinhardt domains whose boundaries are endowed with a general class of smooth measures. We do so by relating it to a Bergman kernel over projective space.

We construct a polynomial of degree 5 from the associative nucleus (kernel) of the free alternative algebra. We show that this polynomial is of minimal degree. Using this polynomial, we obtain decompositions of the varieties of alternative and Malcev algebras.

Let L be a number field containing the pth primitive root of unity ζ p . We investigate the p-rank of the ideal class groups of some subfields of L by using reflection theorems and establish relations between the p-rank of the ideal class groups and that of groups of units of some subfields of L. Let F be a number field and 𝒪 F the ring of integers in F. We also study the p-rank of tame kernels of F and establish relations between the p-rank of K 2𝒪 F and that of some direct summands of the ideal class group of F(ζ p ).

We consider the perfect cotorsion pair (𝒫1, 𝒟) over a commutative ring (often over a domain) R consisting of modules of projective dimension ≤1 and of divisible modules, respectively. The kernel modules in this cotorsion pair are well kown (Theorem 2.1), and those domains are identified whose injectives are kernel modules (Theorem 2.3). The kernel modules always generate the cotorsion pair (𝒫1, 𝒟); they also cogenerate it if the global dimension of the domain R is finite (Theorems 3.1, 3.2). An analogue of the well-known Faith–Walker theorem on injective modules is proved: an integral domain R is noetherian of Krull dimension 1 if there is a cardinal number λ such that every weak-injective (or divisible) R-module is a direct sum of modules of cardinalities ≤ λ (Theorem 4.4). The proof relies on our Theorem 4.1 which generalizes the Faith–Walker theorem as well as a theorem by Guil and Herzog in [14]. Theorem 4.2 provides a general sufficient criterion for a complete cotorsion pair to be Σ-cotorsion; here ...

We relate the existence and regularity of a solution operator to on smoothly bounded pseudoconvex domains to the existence and regularity of a projection operator onto the kernel of .

A class 𝒳 of abelian p-groups is closed under ω1-bijective homomorphisms if whenever f: G → H is a homomorphism with countable kernel and cokernel, then G ∈ 𝒳 iff H ∈ 𝒳. For an ordinal α, we consider the smallest class with this property containing (a) the p α-bounded simply presented groups; (b) the p α-projective groups; (c) the subgroups of p α-bounded simply presented groups. This builds upon classical results of Nunke from [14] and [15]. Particular attention is paid to the separable groups in these classes.

Let X and Y be two infinite dimensional real or complex Banach spaces, and let φ: ℒ(X) → ℒ(Y) be an additive surjective mapping that preserves semi-Fredholm operators in both directions. In the complex Hilbert space context, Mbekhta and Semrl [M. Mbekhta and P. Semrl, Linear maps preserving semi-Fredholm operators and generalized invertibility, Linear Multilinear Algebra 57 (2009), pp. 55–64] determined the structure of the induced map on the Calkin algebra. In this article, we show the following: given an integer n ≥ 1, if φ preserves in both directions ℳ n (X) (resp., 𝒬 n (X)), the set of semi-Fredholm operators on X of non-positive (resp., non-negative) index, having dimension of the kernel (resp., codimension of the range) less than n, then φ(T) = UTV for all T or φ(T) = UT*V for all T, where U and V are two bijective bounded linear, or conjugate linear, mappings between suitable spaces.

The combinatorial rank of the multiparameter version of the Lusztig small quantum group u q (𝔰𝔬2n+1) (the so called Frobenius–Lusztig kernel of type B n ) equals ⌊log 2(n − 1)⌋ +2 provided that q has a finite multiplicative order t > 4.

We consider a semi-classical Dirac operator in arbitrary spatial dimensions with a smooth potential whose partial derivatives of any order are bounded by suitable constants. We prove that the distribution kernel of the inverse operator evaluated at two distinct points fulfilling a certain hypothesis can be represented as the product of an exponentially decaying factor involving an associated Agmon distance and some amplitude admitting a complete asymptotic expansion in powers of the semi-classical parameter. Moreover, we find an explicit formula for the leading term in that expansion., 46 pages

In this paper, we prove that the quantum Gross Laplacian and the quantum Beltrami Laplacian denoted, respectively, as and are rotation-invariant operators. For this purpose, we use the Schwartz–Grothendieck kernel theorem and the characterization theorem of rotation-invariant distributions and operators. Then, we give a characterization of all quantum operators by means of rotation invariance.

Let R be a prime ring with left Martindale quotient ring R ℱ and symmetric Martindale quotient ring Q. Define, for an automorphism σ of R, R (σ) = {x ∈ R∣x σ = x}. Let σ and τ be automorphisms of R, and assume that σ is left R ℱ -algebraic. We show that R (σ) ⊆ R (τ) if and only if x τ = vx σ i v −1 for all x ∈ R, where i is an integer and where v is in the centralizer of R (σ) in Q.

Let U n be a U-statistic based on a symmetric kernel h(x, y) and i.i.d. samples {X, X n ; n ≥ 1}. In this article, the exact moment convergence rates in the first moment of U n are obtained, which extend previous results concerning partial sums.

In this paper we consider the Laplace-Beltrami operator \Delta on Damek-Ricci spaces and derive pointwise estimates for the kernel of exp(\tau \Delta), when \tau \in C* with Re(\tau) \geq 0. When \tau \in iR*, we obtain in particular pointwise estimates of the Schr\"odinger kernel associated with \Delta. We then prove Strichartz estimates for the Schr\"odinger equation, for a family of admissible pairs which is larger than in the Euclidean case. This extends the results obtained by Anker and Pierfelice on real hyperbolic spaces. As a further application, we study the dispersive properties of the Schr\"odinger equation associated with a distinguished Laplacian on Damek-Ricci spaces, showing that in this case the standard dispersive estimate fails while suitable weighted Strichartz estimates hold.

In this article, we study the asymptotic behaviour of solutions to the nonlocal operator u t (x, t) = (−1) n−1 (J * Id − 1) n (u(x, t)), x ∈ ℝ N , which is the nonlocal analogous to the higher order local evolution equation v t = (−1) n−1(Δ) n v. We prove that the solutions of the nonlocal problem converge to the solution of the higher order problem with the right-hand side given by powers of the Laplacian when the kernel J is rescaled in an appropriate way. Moreover, we prove that solutions to both equations have the same asymptotic decay rate as t goes to infinity.

We consider on ℝ the differential-difference operator which is referred to as the Dunkl operator with parameter γ+1/2 associated with the reflection ℤ2 on ℝ. We exhibit a Sonine-type integral representation for the Dunkl kernel e γ, eigenfunction of the Dunkl operator Λγ. This enables us to construct a new Sonine integral transform on ℝ tied to Λγ, which turns out together with its dual to be transmutation operators between two Dunkl operators of different parameters.

Suppose $V$ is a vector space with ${\rm dim} V=p\geq q\geq\aleph_0$, and let $T(V)$ denote the semigroup (under composition) of all linear transformations of $V$. For each $\alpha\in T(V)$, let ${\rm ker}\alpha$ and ${\rm ran}\alpha$ denote the `kernel' and the `range' of $\alpha$, and write $n(\alpha)={\rm dim}{\rm ker}\alpha$ and $d(\alpha)={\rm codim}{\rm ran}\alpha$. In this paper, we study the semigroups $AM(p,q) =\{\alpha\in T(V):n(\alpha), Fundação para a Ciência e a Tecnologia (FCT)

In this article for C 1-functions f, given in the matrix domain integral representations of the form are obtained. Here P is the orthogonal projector of the space L 2{R mn ; [det(I m − ηη*)]αdμ mn (η)} onto its subspace of holomorphic functions and the integral operator T has an explicit kernel with estimates.

Let K be the fraction field of a Henselian discrete valuation ring. In Bazyleu et al. (2007) an explicit description of the central simple algebras of exponent 2 in the kernel of the natural map 2 Br(K(x)) → ∏ω∈Ω 2 Br(K(x)ω), (the product taken over all orderings ω and K(x)ω the real closure of K(x) at ω), is given. This allows to describe a class of conic bundle surfaces over K in terms of their local data (i.e., in terms of their degenerate fibres). Such a description is given in Theorem 3.1, the main result of this note.

An endomorphism on an algebra 𝒜 is said to be “strong” if it is compatible with every congruence on 𝒜; and 𝒜 is said to have the “strong endomorphism kernel property” if every congruence on 𝒜, different from the universal congruence, is the kernel of a strong endomorphism on 𝒜. Here we consider this property in the context of Ockham algebras. In particular, for those MS-algebras that have this property we describe the structure of their dual space in terms of 1-point compactifications of discrete spaces.

We give a simple proof of the Kernel theorem for the spaces of tempered ultradistributions of Beurling–Komatsu and of Roumieu–Komatsu types, by using the characterization of Fourier–Hermite coefficients of the elements of the spaces. As a consequence of the Kernel theorem, we have that the Weyl transform can be extended on spaces of tempered ultradistributions.

In this article, choosing a proper integral transformation concerning the unknown function, by means of interpolation method and the extended residue theorem, on the basis of analysing the internal relations among the necessary conditions of solvability and with the help of discussing the smooth properties of difference quotient functions, we obtain the solutions with singularities of high order and the solvable conditions for a class of systems of complete singular integral equations with coefficients and kernel densities possessing certain analytic properties. The case of a single equation, the case of solutions with singularities of order 1 or solutions in class H, etc., may all be regarded as the special cases. Dedicated to Professor Lu Jianke on the occasion of his 85th birthday.

A class of pseudo-differential operators (p.d.o.'s) associated with the general Fourier kernel studied by Hardy and Titchmarsh is defined. A symbol class T m is introduced. It is shown that the p.d.o.'s associated with the symbol are continuous linear mappings of the Braaksma and Schuitman space T(λ,μ) into itself. An integral representation of p.d.o. is obtained. Some special forms of the symbol are considered. It is shown that these p.d.o.'s and their products are bounded in certain Sobolev type space.

We characterize the Jordan domain of a given perimeter L > 4 such that the two designated interior points ±1 lying in its interior have the smallest hyperbolic distance between them. We also show how this result serves as an example illustrating the theory of quadrature domains studied by Bjorn Gustafsson and Harold Shapiro. †Dedicated to Professor Peter L. Duren on the occasion of his 70th birthday.

Let R denote a commutative ring with identity. Let A ∈ M n×n (R). The kernel of the natural map ϑ A : R[x] → M n×n (R) given by ϑ A (a t x t + ··· + a 1 x + a 0) = a t A t + ··· + a 1 A + a 0 I n is denoted by N A and called the null ideal of A. In this article, the question of when N A is a principal ideal in R[x] is studied. For n = 2, N A is principal for every matrix A ∈ M 2 × 2 (R) if and only if R is a P.P. ring, i.e., every principal ideal in R is projective. If R contains only finitely many minimal primes, then N A is principal for every A ∈ M 2×2(R) if and only if R is a finite product of integral domains. If R contains only finitely many minimal primes, then N A is principal for all n ≥ 1 and for all A ∈ M n×n (R) if and only if R is a finite product of normal domains. For fixed n ≥ 1 and A ∈ M n×n (R), conditions on various R-submodules of N A are given which imply N A is principal.

A bounded linear map Θ defined on an L 2-space with values in a reproducing kernel Hilbert space is necessarily given as an integral operator with kernel K Θ. We discuss in general how the adjoint of K Θ is the generating function associated to a basis of the domain space. Our primary applications are the highest weight representations for a Hermitian group G modeled by the geometric realization. We obtain new formulas relating the generating function to the action of an Abelian subalgebra 𝔭 −⊂𝔤ℂ, where 𝔤 ℂ is the complexification of the Lie algebra of G.

The article is devoted to the study of an asymptotic behavior of the integral transform involving the Whittaker function W ρ, γ(z) in the kernel. It is proved that has power or power-logarithmic asymptotic expansion, as λ→0+ and λ→+∞, provided that f(t) has power asymptotic behavior at infinity and zero, respectively.

An algebra 𝒜 has the endomorphism kernel property if every congruence on 𝒜 different from the universal congruence is the kernel of an endomorphism on 𝒜. We first consider this property when 𝒜 is a finite distributive lattice, and show that it holds if and only if 𝒜 is a cartesian product of chains. We then consider the case where 𝒜 is an Ockham algebra, and describe in particular the structure of the finite de Morgan algebras that have this property.

In this article we explore the semigroup of kernel functors associated to a generalized discrete valuation ring. We will show that, among other interesting features, this semigroup is linearly ordered under divisibility, and we will present a factorization of kernel functors in terms of irreducible elements. The asymmetry of the divisibility condition will also be considered. In a way, these results are closely related to (and in the spirit of) the corresponding results obtained by Brungs for the lattice of ideals of this type of rings.

Sjogren and others studied the boundary behavior of fractional Poisson integrals with respect to the fractional power of the Poisson kernel. We extend the fractional power of the Poisson kernel to a non-integrable kernel and investigate the boundary behavior of associated Poisson integrals. The existence of certain tangential limit (Fatou type theorem) as well as its sharpness (Littlewood type theorem) are given. The admissible tangency varies according to the integrability of the boundary function. Our Littlewood type theorem is new even for the fractional power of the Poisson kernel.

Let ν be a (discrete) valuation of the rational function field k(t), where k is a field, and let 𝒞ν be the intersection of the valuation rings of all the valuations of k(t), other than that of ν. It is well-known that either ν = νπ, the valuation determined by some irreducible polynomial π in k[t], or ν = ν∞, the “valuation at infinity”. In this paper we prove that GL 2(𝒞ν), where ν = νπ, is the fundamental group of a certain tree of groups. The tree has finitely many vertices and its terminal vertices correspond with the elements of the ideal class group of 𝒞ν. This extends a previous result of Nagao for the special case ν = ν∞. In this case 𝒞ν = k[t] and Nagao proves that GL 2(k[t]) is an amalgamated product of a pair of groups. As a consequence we show that, when the degree of π is at least 4, GL 2(𝒞ν) has a free, non-cyclic quotient whose kernel contains (for example) all the unipotent matrices. This represents a two-dimensional anomaly.

The singularities of harmonic functions generated under the action of an integral transform with a fractionalized kernel are studied. A simple relationship arises connecting pairs of singularities.

We study the large time behavior of the solution u to an initial and boundary value problem related to the following integro-differential equation $$ u_{tt} = G_0 \Delta u + \int_0^t G'(t-s) \Delta u(x, s)\, ds - a u_t \eqno(0.1) $$ where G 0 , a are real constant coefficients, G 0 > 0, a S 0 and $ G\,' \in L^1({{\shadR}}^ + ) \cap L^2({{\shadR}}^ + ), G\,' \le 0 $ . It is known that, when G ' L 0 and a > 0, the solution u of (0.1) exponentially decays. Here we prove that, for any nonnegative a and for any $ G ' \not \equiv 0 $ , the solution u of the Eq. (0.1) exponentially decays only if the relaxation kernel G ' does. In other words, the introduction of the dissipative term related to G ' does not allow the exponential decay due to the presence of the positive coefficient a . We also prove analogous results for the polynomial decay.

All known hypergroups on [0, + \infty[ are associated to second order differential operators on ]0, +\infty[ of Sturm-Liouville type. It has been recognized as a crucial problem to determine if every hypergroup on [0, +\infty[ is of this type. In this paper we give an answer to this question. Moreover we show that a Laplace representation formula with non negative kernel always holds for the characters of the hypergroup.

Let S be a regular semigroup and Con S the congruence lattice of S. For every ρ e Con S there exist a greatest congruence ρ K [ρT] and a smallest congruence ρ k [ρt] on S with the same kernel [trace] as ρ. The subsemigroup Γ (S) of the transformation semigroup on Con S generated by the transformations ρ → ρK, ρ → ρk, ρ → ρT, and ρ → ρt, ρ e Con S, is investigated for E-unitary completely regular semigroups. It is shown that in this case Γ(S′) contains 25 elements at most. A 25-element semigroup Γ(S′) is realized for some E-unitary regular orthogroup S′.

We use the results on the minimal basis of the centre of an Iwahori–Hecke algebra from our earlier work, as well as some additional results on the minimal basis, to describe the image and kernel of the Brauer homomorphism for Iwahori–Hecke algebras defined by L. Jones (Jones, L. Centres of Generic Hecke Algebras; Ph.D. Thesis; University of Virginia, 1987.). *The research in this paper was done as part of the work towards a Ph.D. at the University of New South Wales. Several revisions were done while visiting the University of Virginia between 1998 and 2000.

Let S be a completely simple semigroup represented as a Rees matrix semigroup M(I,G,P) with normalized sandwich matrix P. On the congruence lattice C(S) of S we consider the relations T i, K and T r which identify congruences with the same left trace, kernel and right trace, respectively. These are equivalences whose classes are intervals. The upper and lower ends of these intervals induce the following operators on C(S) Tl, K, Tr, tl, k and tr .We construct here the semigroup generated by these operators as a homomorphic image of a semigroup given by generators and relations and demonstrate the minimality of the latter.