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Let A be an n × n matrix. The Hermitian parts of A are denoted by ℜ ( A ) = ( A + A ⁎ ) / 2 and ℑ ( A ) = ( A − A ⁎ ) / ( 2 i ) . The kernel vectors of the linear pencil x ℜ ( A ) + y ℑ ( A ) + z I n play a role for the inverse numerical range of A. This kernel vector technique was applied to perform the inverse numerical range of 3 × 3 symmetric matrices. In this paper, we follow the kernel vector method and apply the Abel theorem for 3 × 3 Hermitian matrices. We present the elliptic curve group structure of the cubic curve associated to the ternary form of the matrix, and characterize the Abel type additive structure of the divisors of the cubic curve. A numerical example is given to illustrate the characterization related to the Riemann theta representation.

The goal of this article is to study some properties of generalized M−series introduced by Sharma and Jain in 2009. Here, we establish two theorems which provides the images of product of Srivastava’s polynomials and this M−series under the generalized fractional integral operators involving Fox’s H-function as kernel. Corresponding assertions in terms of Saigo, ErdeIyi-Kober, Riemann-Liouville and Weyl type of fractional integrals are also presented. In the concluding part, a variety of well-known special cases are also mentioned.

In this article, we study the fundamental solution of the operator $$\left((\diamond+m^2)\left(\frac{\triangle^2+\boxdot^2}{2}\right)\right)^{k}$$ , iterated $k$ -times and is defined by ( \ref { odot }) ,where $m$ is a nonnegative real number, and $k$ is a nonnegative integer. After that, we study the Fourier transform of the operator $ \left((\diamond+m^2)\left(\frac{\triangle^2+\boxdot^2}{2}\right)\right)^{k}\delta$ .

This paper is dedicated to L p bounds on eigenfunctions of a Schrodinger-type operator ( − Δ g ) α / 2 + V on closed Riemannian manifolds for critically singular potentials V. The operator ( − Δ g ) α / 2 is defined spectrally in terms of the eigenfunctions of − Δ g . We obtain also quasimodes and spectral clusters estimates. As an application, we derive Strichartz estimates for the fractional wave equation ( ∂ t 2 + ( − Δ g ) α / 2 + V ) u = 0 . The wave kernel techniques recently developed by Bourgain-Shao-Sogge-Yao [4] and Shao-Yao [27] play a key role in this paper. We construct a new reproducing operator with several local operators and some good error terms. Moreover, we shall prove that these local operators satisfy certain variable coefficient versions of the “uniform Sobolev estimates” by Kenig-Ruiz-Sogge [18] . These enable us to handle the critically singular potentials V and prove the quasimode estimates.

This paper focuses on constructions of MDS self-dual codes from (extended) generalized Reed-Solomon (GRS) codes. Let $q = r^{2}$ be an odd prime power. We show that, there exists a $q$ -ary self-dual (extended) GRS code for each even length in the range $[{2r,3r-3}]$ , and for each singly even length in the range $[3r-1,4r]$ . This extends the only known consecutive range $[2,2r]$ to $[{2,3r-3}]$ for this case. Furthermore, our general constructions provide many MDS self-dual codes with new parameters which, to the best of our knowledge, were not reported before.

In the present paper, we study a purely inseparable counterpart of Abhyankar’s conjecture for the affine line in positive characteristic, and prove its validity for all the finite local non-abelian simple group schemes in characteristic $$p>5$$ . The crucial point is how to deal with finite local group schemes which cannot be realized as the Frobenius kernel of a smooth algebraic group. Such group schemes appear as the ones associated with Cartan type Lie algebras. We settle the problem for such Lie algebras by making use of natural gradations or filtrations on them.

We extend to arbitrary commutative base rings a recent result of Demeneghi that every ideal of an ample groupoid algebra over a field is an intersection of kernels of induced representations from isotropy groups, with a much shorter proof, by using the author's Disintegration Theorem for groupoid representations. We also prove that every primitive ideal is the kernel of an induced representation from an isotropy group; however, we are unable to show, in general, that it is the kernel of an irreducible induced representation. If each isotropy group is finite (e.g., if the groupoid is principal) and if the base ring is Artinian (e.g., a field), then we can show that every primitive ideal is the kernel of an irreducible representation induced from isotropy.

Let A be an n-by-n matrix and M ( x , y , z ) = z I n + x ℜ ( A ) + y ℑ ( A ) , where ℜ ( A ) = ( A + A ⁎ ) / 2 and ℑ ( A ) = ( A − A ⁎ ) / ( 2 i ) . The inverse numerical range problem seeks a unit vector x corresponding to a given point z of the numerical range of A satisfying z = x ⁎ A x . A kernel vector function ξ = ξ ( x , y , z ) of M ( x , y , z ) with point ( x , y , z ) on the curve F A ( x , y , z ) = det ⁡ ( M ( x , y , z ) ) = 0 plays the role of the unit vector x for the inverse numerical range. The columns of the adjugate matrix L ( x , y , z ) = ( L j k ( x , y , z ) ) of M ( x , y , z ) are kernel vector functions of M ( x , y , z ) . We prove the Abel theorem on the intersections of the algebraic curves F A ( x , y , z ) = 0 and L j k ( x , y , z ) = 0 . A concrete numerical example is provided to verify the result using the Maple package algcurves.

This work studies the kernel of a linear operator associated with the generalized k-fold commutator. Given a set $\mathfrak{A}= \left\{ A_{1}, \ldots ,A_{k} \right\}$ of real $n \times n$ matrices, the commutator is denoted by$[A_{1}| \ldots |A_{k}]$. For a fixed set of matrices $\mathfrak{A}$ we introduce a multilinear skew-symmetric linear operator $T_{\mathfrak{A}}(X)=T(A_{1}, \ldots ,A_{k})[X]=[A_{1}| \ldots |A_{k} |X] $. For fixed $n$ and $k \ge 2n-1, \; T_{\mathfrak{A}} \equiv 0$ by the Amitsur--Levitski Theorem [2] , which motivated this work. The matrix representation $M$ of the linear transformation $T$ is called the k-commutator matrix. $M$ has interesting properties, e.g., it is a commutator; for $k$ odd, there is a permutation of the rows of $M$ that makes it skew-symmetric. For both $k$ and $n$ odd, a provocative matrix $\mathcal{S}$ appears in the kernel of $T$. By using the Moore--Penrose inverse and introducing a conjecture about the rank of $M$, the entries of $\mathcal{S}$ are shown to be quotients of polynomials in the entries of the matrices in $\mathfrak{A}$. One case of the conjecture has been recently proven by Brassil. The Moore--Penrose inverse provides a full rank decomposition of $M$.

The congruence subgroup problem for a finitely generated group Γ and for G ≤ Aut ⁢ ( Γ ) G\leq\mathrm{Aut}(\Gamma) asks whether the map G ^ → Aut ⁢ ( Γ ^ ) \hat{G}\to\mathrm{Aut}(\hat{\Gamma}) is injective, or more generally, what its kernel C ⁢ ( G , Γ ) C(G,\Gamma) is. Here X ^ \hat{X} denotes the profinite completion of 𝑋. In the case G = Aut ⁢ ( Γ ) G=\mathrm{Aut}(\Gamma) , we write C ⁢ ( Γ ) = C ⁢ ( Aut ⁢ ( Γ ) , Γ ) C(\Gamma)=C(\mathrm{Aut}(\Gamma),\Gamma) . Let Γ be a finitely generated group, Γ ¯ = Γ / [ Γ , Γ ] \bar{\Gamma}=\Gamma/[\Gamma,\Gamma] , and Γ * = Γ ¯ / tor ⁢ ( Γ ¯ ) ≅ Z ( d ) \Gamma^{*}=\bar{\Gamma}/\mathrm{tor}(\bar{\Gamma})\cong\mathbb{Z}^{(d)} . Define Aut * ⁢ ( Γ ) = Im ⁡ ( Aut ⁢ ( Γ ) → Aut ⁢ ( Γ * ) ) ≤ GL d ⁢ ( Z ) . \mathrm{Aut}^{*}(\Gamma)=\operatorname{Im}(\mathrm{Aut}(\Gamma)\to\mathrm{Aut}(\Gamma^{*}))\leq\mathrm{GL}_{d}(\mathbb{Z}). In this paper we show that, when Γ is nilpotent, there is a canonical isomorphism C ⁢ ( Γ ) ≃ C ⁢ ( Aut * ⁢ ( Γ ) , Γ * ) . C(\Gamma)\simeq C(\mathrm{Aut}^{*}(\Gamma),\Gamma^{*}). In other words, C ⁢ ( Γ ) C(\Gamma) is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group Aut * ⁢ ( Γ ) \mathrm{Aut}^{*}(\Gamma) . In particular, in the case where Γ = Ψ n , c \Gamma=\Psi_{n,c} is a finitely generated free nilpotent group of class 𝑐 on 𝑛 elements, we get that C ⁢ ( Ψ n , c ) = C ⁢ ( Z ( n ) ) = { e } C(\Psi_{n,c})=C(\mathbb{Z}^{(n)})=\{e\} whenever n ≥ 3 n\geq 3 , and C ⁢ ( Ψ 2 , c ) = C ⁢ ( Z ( 2 ) ) = F ^ ω C(\Psi_{2,c})=C(\mathbb{Z}^{(2)})=\hat{F}_{\omega} is the free profinite group on countable number of generators.

Given a monomial $m$ in a polynomial ring and a subset $L$ of the variables of the polynomial ring, the principal $L$-Borel ideal generated by $m$ is the ideal generated by all monomials which can be obtained from $m$ by successively replacing variables of $m$ by those which are in $L$ and have smaller index. Given a collection $\mathcal{I}=\{I_1,\ldots,I_r\}$ where $I_i$ is $L_i$-Borel for $i=1,\ldots,r$ (where the subsets $L_1,\ldots,L_r$ may be different for each ideal), we prove in essence that if the bipartite incidence graph among the subsets $L_1,\ldots,L_r$ is chordal bipartite, then the defining equations of the multi-Rees algebra of $\mathcal{I}$ has a Gr\"obner basis of quadrics with squarefree lead terms under lexicographic order. Thus the multi-Rees algebra of such a collection of ideals is Koszul, Cohen-Macaulay, and normal. This significantly generalizes a theorem of Ohsugi and Hibi on Koszul bipartite graphs. As a corollary we obtain that the multi-Rees algebra of a collection of principal Borel ideals is Koszul. To prove our main result we use a fiber-wise Gr\"obner basis criterion for the kernel of a toric map and we introduce a modification of Sturmfels' sorting algorithm., Comment: 29 pages, 4 figures

In this work, we present the notion of a multiplier on AT-algebra and investigate several properties. Also, some theorems and examples are discussed. The notions of the kernel and the image of multipliers are defined. After that, some propositions related to isotone and regular multipliers are proved. Finally, the Left and the Right derivations of the multiplier are obtained

Let G be a finite group, p be a prime divisor of |G|, and P be a Sylow p-subgroup of G. We prove that P is normal in a solvable group G if |G: ker φ|p′ = φ(1)p′ for every nonlinear irreducible monomial p-Brauer character φ of G, where ker φ is the kernel of φ and φ(1)p′ is the p′-part of φ(1).

Let L be a globally generated line bundle over a smooth irreducible complex projective surface X. The syzygy bundle $$M_{L}$$ is the kernel of the evaluation map $$H^0(L)\otimes \mathcal O_X\rightarrow L$$ . We prove the L-stability of $$M_L$$ for Hirzebruch surfaces, del Pezzo surfaces and Enriques surfaces. The $$(-K_X)$$ -stability of syzygy bundles $$M_L$$ over del Pezzo surfaces is also obtained.

Let $\Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{d-1}$, $T_{\Omega}$ be the homogeneous singular integral operator with kernel $\frac{\Omega(x)}{|x|^d}$ and $T_{\Omega,\,b}$ be the commutator of $T_{\Omega}$ with symbol $b$. In this paper, we prove that if $\Omega\in L(\log L)^2(S^{d-1})$, then for $b\in {\rm BMO}(\mathbb{R}^d)$, $T_{\Omega,\,b}$ satisfies an endpoint estimate of $L\log L$ type., Comment: We reorganized the manuscript

The characterization of the points in C p : 1 ≤ p < ∞ ( ℋ ) {C}_{p{:}_{1\le p\lt \infty }}\left({\mathcal{ {\mathcal H} }}) , the Von Neuman-Schatten p-classes, that are orthogonal to the range of elementary operators has been done for certain kinds of elementary operators. In this paper, we shall study this problem of characterization on an abstract reflexive, smooth and strictly convex Banach space for arbitrary operator. As an application, we consider other kinds of elementary operators defined on the spaces C p : 1 ≤ p < ∞ ( ℋ ) {C}_{p{:}_{1\le p\lt \infty }}\left({\mathcal{ {\mathcal H} }}) , and finally, we give a counterexample to Mecheri’s result given in this context.

In this paper, we study the existence of solutions for a generalized sequential Caputo-type fractional neutral differential inclusion with generalized integral conditions. The used fractional operator has the generalized kernel in the format of $( \vartheta (t)-\vartheta (s)) $ ( ϑ ( t ) − ϑ ( s ) ) along with differential operator $\frac{1}{\vartheta '(t)}\,\frac{\mathrm{d}}{\mathrm{d}t}$ 1 ϑ ′ ( t ) d d t . We obtain existence results for two cases of convex-valued and nonconvex-valued multifunctions in two separated sections. We derive our findings by means of the fixed point principles in the context of the set-valued analysis. We give two suitable examples to validate the theoretical results.

The structural determination of alloy hydride nanoclusters with high nuclearity remains challenging. We herein report the synthetic procedure and the structural elucidation of an Au-Ag alloy nanocluster with 12 hydride ligands-[Au16Ag43H12(SPhCl2)34]5-. The structure of [Au16Ag43H12(SPhCl2)34]5- comprises an Au16Ag3 kernel that is stabilized by 12 hydride ligands, 8 thiol bridges, and 6 Agm(SR)n motif units. The 12 hydride ligands in Au16Ag43 have been confirmed by both 2H NMR and ESI-MS measurements, and their positions have been theoretically evaluated, located at the interlayer between the Au16Ag3 kernel and the Ag-SR shell. The metastable [Au16Ag43H12(SPhCl2)34]5- can convert to [Au12Ag32(SPhCl2)30]4- spontaneously. Structurally, the Au16Ag3 kernel of [Au16Ag43H12(SPhCl2)34]5- could be regarded as the overlapping of two hollow Au8Ag3 cages via sharing an Ag3 line, which is in contrast to the solely icosahedral Au12 kernel of [Au12Ag32(SPhCl2)30]4-. Besides, the overall construction of Au16Ag43 or Au12Ag32 follows a complementing or overlapping assembly mode, respectively. Overall, the structural anatomy of Au16Ag43H12(SPhCl2)34 sheds some new insight into the structural evolution of metal nanoclusters.

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 ...

In this paper we study fractional initial value problems with Caputo–Fabrizio derivative which involves nonsingular kernel. First we apply α-ℓ-contraction and α-type F-contraction mappings to study the existence and uniqueness of solutions for such problems. Finally, we use some contraction mappings in complete $\mathfrak{F}$ F -metric spaces for this purpose.

The goal of this paper is to prove theorems that elucidate the Cohen-Lenstra-Martinet conjectures for the distributions of class groups of number fields, and further the understanding of their implications. We start by giving a simpler statement of the conjectures. We show that the probabilities that arise are inversely proportional the to number of automorphisms of structures slightly larger than the class groups. We find the moments of the Cohen-Lenstra-Martinet distributions and prove that the distributions are determined by their moments. In order to apply these conjectures to class groups of non-Galois fields, we prove a new theorem on the capitulation kernel (of ideal classes that become trivial in a larger field) to relate the class groups of non-Galois fields to the class groups of Galois fields. We then construct an integral model of the Hecke algebra of a finite group, show that it acts naturally on class groups of non-Galois fields, and prove that the Cohen-Lenstra-Martinet conjectures predict a distribution for class groups of non-Galois fields that involves the inverse of the number of automorphisms of the class group as a Hecke-module.

In this paper, we study fuzzy congruence relations and kernel fuzzy ideals of an Ockham algebra A , f , whose truth values are in a complete lattice satisfying the infinite meet distributive law. Some equivalent conditions are derived for a fuzzy ideal of an Ockham algebra A to become a fuzzy kernel ideal. We also obtain the smallest (respectively, the largest) fuzzy congruence on A having a given fuzzy ideal as its kernel.

The main aim of this paper is to describe the closure of a finitely generated subgroup of a finitely generated free group in the proabelian topology. Our approach depends heavily on the description of the abelian kernel of a finite inverse semigroup.

Let $G$ be a finite group and $k$ an algebraically closed field of characteristic $p>0$. We define the notion of a Dade $kG$-module as a generalization of endo-permutation modules for $p$-groups. We show that under a suitable equivalence relation, the set of equivalence classes of Dade $kG$-modules forms a group under tensor product, and the group obtained this way is isomorphic to the Dade group $D(G)$ defined by Lassueur. We also consider the subgroup $D^{\Omega} (G)$ of $D(G)$ generated by relative syzygies $\Omega_X$, where $X$ is a finite $G$-set. If $C(G,p)$ denotes the group of superclass functions defined on the $p$-subgroups of $G$, there are natural generators $\omega_X$ of $C(G,p)$, and we prove the existence of a well-defined group homomorphism $\Psi_G:C(G,p)\to D^\Omega(G)$ that sends $\omega_X$ to $\Omega_X$. The main theorem of the paper is the verification that the subgroup of $C(G,p)$ consisting of the dimension functions of $k$-orientable real representations of $G$ lies in the kernel of $\Psi_G$., Comment: Minor revision, 40 pages

We consider a nonlinear integro-differential equation with zero operator in the differential part whose integral operator contains several rapidly varying kernels. This paper continues the research carried out earlier for equations with only one rapidly varying kernel. We prove that the conditions for the solvability of the corresponding iteration problems, as in the linear case, are not differential (as in problems with nonzero operator in the differential part) but rather integro-differential equations, and the structure of these equations is substantially influenced by the nonlinearity. In the nonlinear case, so-called resonances can arise, significantly complicating the development of the corresponding algorithm of the regularization method. The paper deals with the nonresonant case.

The classes of band-dominated operators and the subclass of operators in the Wiener algebra $${\mathcal {W}}$$ W are known to be inverse closed. This paper studies and extends partially known results of that type for one-sided and generalized invertibility. Furthermore, for the operators in the Wiener algebra $${\mathcal {W}}$$ W invertibility, the Fredholm property and the Fredholm index are known to be independent of the underlying space $$l^p$$ l p , $$1\le p\le \infty $$ 1 ≤ p ≤ ∞ . Here this is completed by the observation that even the kernel and a suitable direct complement of the range as well as generalized inverses of operators in $${\mathcal {W}}$$ W are invariant w.r.t. p.

The purpose of this paper is to find pointwise estimates on the integral kernel for the $$\Box _b$$ Green operator, N, for a particular quadric submanifold of codimension 2 in $${\mathbb {C}}^4$$ . This particular kernel has singularities away from the diagonal. Furthermore, the estimates exhibited by this kernel, both on and off the diagonal, follow no known paradigm.

In this paper, we investigate the existence of a local solution in time and discuss the exponential asymptotic behavior to a weakly damped wave equation involving the variable-exponents $$ \begin{array}{@{}rcl@{}} &&u_{tt}-M\left( \left\vert \nabla u\left( t\right) \right\vert^{2}\right) {\Delta} u+{{\int}_{0}^{t}}g\left( t-s\right) {\Delta} u\left( s\right) ds+\gamma_{1}u_{t}+\left\vert u_{t}\right\vert^{k\left( x\right) -1}u_{t}\\ &=&\left\vert u\right\vert^{p\left( x\right) -1}u \text{ in }{\Omega} \times \mathbb{R}^{+} \end{array} $$ with simply supported boundary condition, where Ω is a bounded domain of $\mathbb {R}^{n}$ , g > 0 is a memory kernel that decays exponentially, and M(s) is a locally Lipschitz function. This kind of problem without the memory term when k(.) and p(.) are constants models viscoelastic Kirchhoff equation.

Let G be a finite Frobenius group of degree n. We show, by elementary means, that n is a power of some prime p provided the rank $${\mathrm{rk}}(G)\le 3+\sqrt{n+1}$$ rk ( G ) ≤ 3 + n + 1 . Then the Frobenius kernel of G agrees with the (unique) Sylow p-subgroup of G. So our result implies the celebrated theorems of Frobenius and Thompson in a special situation.

An absolute parallelism for $2$-nondegenerate CR manifolds $M$ of hypersurface type was recently constructed independently by Isaev-Zaitsev, Medori-Spiro, and Pocchiola in the minimal possible dimension ($\dim M=5$), and for $\dim M=7$ in certain cases by the first author. We develop a bigraded analog of Tanaka's prolongation procedure to construct a canonical absolute parallelism for these CR structures in arbitrary (odd) dimension with Levi kernel of arbitrary admissible dimension. We introduce the notion of a bigraded Tanaka symbol. Under regularity assumption that the symbol is a Lie algebra, we define a bigraded analog of the Tanaka universal algebraic prolongation and prove that for any CR structure with a given regular symbol there exists a canonical absolute parallelism on a bundle whose dimension is that of this bigraded prolongation. We show that there is a unique (up to local equivalence) such CR structure whose algebra of infinitesimal symmetries has maximal possible dimension, and the latter algebra is isomorphic to the real part of the bigraded prolongation of the symbol. In the case of $1$-dimensional Levi kernel we classify all regular symbols and calculate their bigraded prolongations. In this case the regular symbols can be subdivided into nilpotent, strongly non-nilpotent and weakly non-nilpotent. The bigraded prolongation of strongly non-nilpotent symbols is isomorphic to $\mathfrak{so}\left(m,\mathbb C\right)$ where $m=\tfrac{1}{2}(\dim M+5)$. Any real form of this algebra, except $\mathfrak{so}\left(m\right)$ and $\mathfrak{so}\left(m-1,1\right)$, corresponds to the real part of the bigraded prolongation of exactly one strongly non-nilpotent symbol. However, for a fixed $\dim M\geq 7$ the dimension of the bigraded prolongations achieves its maximum on one of the nilpotent regular symbols, and this maximal dimension is equal to $\tfrac{1}{4}(\dim M-1)^2+7$., 44 pages; Accepted to Journal f\"ur die reine und angewandte Mathematik (Crelle's journal); the functorial property of the canonical frame is emphasized, some references were added/updated

We introduce a new class of torsion-free abelian groups which are special epimorphic images of finite rank completely decomposable groups. They belong to the well-known class of Butler groups by their definition. In comparison with the existing results the presented groups are not only of the maximal possible rank that is obviously less than the rank of the pre-image. The kernel of the above epimorphism admits a special matrix representation. The number of matrix rows is equal to the kernel rank and the number of its columns coincides with the preimage rank. By finitely many suitable operations the original matrix representation can be deduced to a special trapezoid form which corresponds to another choice of the epimorphism kernel generators. This matrix form clarifies the properties of the image that is exactly a group under investigation. For this class of torsion-free abelian groups of finite rank a strong indecomposability criterion is proved on the basis of their matrix representation.

We discuss the work of Birman and Solomyak on the singular numbers of integral operators from the point of view of modern approximation theory, in particular, with the use of wavelet techniques. We are able to provide a simple proof of norm estimates for integral operators with kernel in $$B^{1/p-1/2}_{p,p}(\mathbb R,L_2(\mathbb R))$$ . This recovers, extends, and sheds new light on a theorem of Birman and Solomyak. We also use these techniques to provide a simple proof of Schur multiplier bounds for double operator integrals with bounded symbol in $$B^{1/p-1/2}_{2p/(2-p),p}(\mathbb R,L_\infty(\mathbb R))$$ , which extends Birman and Solomyak’s result to symbols without compact domain.

With the use of the theory of generalized inversion of operators and integral operators, we obtain a criterion of solvability and determine the general form of solutions of a linear boundary-value problem for the integrodifferential equation with degenerate kernel in a Banach space.

In the present work, we investigate the global existence and uniform decay rates of solutions to the Cauchy problem in $$\mathbf {R}^{n}$$ ( $$n\ge 1)$$ related to the dynamic behavior of evolution equations accounting for rotational inertial forces along with a linear viscoelastic memory damping arising in viscoelastic materials. Under certain conditions on the initial data and on the kernel, the global existence and decay estimates of the solutions are established. Furthermore, time decay estimates in higher Sobolev space of the solution are provided. The proof is carried out by means of the point-wise decay estimates of the solution in the Fourier space.

In this paper, the first family of conforming discrete three dimensional Gradgrad-complexes consisting of finite element spaces is constructed. These discrete complexes are exact in the sense that the range of each discrete map is the kernel space of the succeeding one. These spaces can be used in the mixed form of the linearized Einstein-Bianchi system.

For an admissible congruence $$\gamma $$ on an ample semigroup S, the minimum admissible congruence $$\gamma _t$$ whose trace is $$\mathrm {tr}\gamma $$ has been determined by the author, and Fountain, Gomes and Gould. With some restriction on $$\gamma $$ , we characterize in this paper the minimum admissible congruence $$(\gamma _t)_k$$ whose kernel is $$\mathrm {ker}\gamma _t$$ , as well as the minimum admissible congruence $$((\gamma _t)_k)_t$$ whose trace is $$\mathrm {tr}(\gamma _t)_k$$ . These results extend the corresponding results of Petrich and Reilly in the inverse semigroup case. Meanwhile, some particular minimum admissible congruences on S are recognized and this information is applied to set up a network of certain minimum admissible congruences on S.

We consider discrete Dirac systems as an alternative (to the famous Szeg\H{o} recurrencies and matrix orthogonal polynomials) approach to the study of the corresponding block Toeplitz matrices. We prove an analog of the Christoffel--Darboux formula and derive the asymptotic relations for the analog of reproducing kernel (using Weyl--Titchmarsh functions of discrete Dirac systems). We study also the case of rational Weyl--Titchmarsh functions (and GBDT version of the B\"acklund-Darboux transformation of the trivial discrete Dirac system). We show that block diagonal plus block semi-separable Toeplitz matrices appear in this case., Comment: This work is an important development of the idea presented in our paper arXiv:1711.03064

In this work for a class of second order model and non-model partial integro-differential equations with singularity in the kernel by the aid of arbitrary functions obtained integral representation for solution manifold. In this given article it is entered a new class of functions that at point (a, b) is converted to zero with some asymptotic behavior and the solution of given equation is found in considering class. Also, singular integro-differential operators are entered and main properties of these operators are learned. Also inverse operators for these operators are found. It is shown that the solution of studying equation is equivalent to the solution of system of two ordinary integro-differential equations of variables x and y. In the cases, when the solution of given integro-differential equation depends of any arbitrary constant Cauchy type problems are investigated. In order to study Cauchy type problems the properties of their solutions are studied. It is shown that, when some conditions are fulfilled the each of Cauchy type problems has only unique solution.

The relationship between a transitive GE-algebra and a belligerent GE-algebra (also, between an antisymmetric GE-algebra and a left exchangeable GE-algebra) is displayed. A condition for the trivial GE-filter to be a belligerent GE-filter is provided. The least GE-filter containing a given GE-filter and one element is formed. Conditions under which any set can be turned into a GE-filter are described. The notions of $$\odot $$ -GE-algebras and prominent GE-filters are introduced, and their properties are investigated. The relationship between a prominent GE-filter and a GE-filter are considered, and conditions for a GE-filter and the trivial GE-filter to be a prominent GE-filter are given. The conditions under which the upset of an element will be a prominent GE-filter are examined, and the extension property for the prominent GE-filter is established. The notion of GE-morphism is introduced, and the GE-morphism theorem is considered. Conditions for the kernel of a GE-morphism to be a belligerent GE-filter are provided, and the condition that if the kernel of a GE-morphism is a belligerent GE-filter, then the trivial GE-filter becomes a belligerent GE-filter is given. A condition for the kernel of a GE-morphism to be a belligerent GE-filter is discussed, and the vice versa is also considered.

Let $S_n$ be the simple random walk on the integer lattice $\mathbb{Z}^d$. For a Bernstein function $\phi$ we consider a random walk $S^\phi_n$ which is subordinated to $S_n$. Under a certain assumption on the behaviour of $\phi$ at zero we establish global estimates for the transition probabilities of the random walk $S^\phi_n$. The main tools that we apply are the parabolic Harnack inequality and appropriate bounds for the transition kernel of the corresponding continuous time random walk.

Let $M = \mathbb{B}^2 / \Gamma$ be a smooth ball quotient of finite volume with first betti number $b_1(M)$ and let $\mathcal{E}(M) \ge 0$ be the number of cusps (i.e., topological ends) of $M$. We study the growth rates that are possible in towers of finite-sheeted coverings of $M$. In particular, $b_1$ and $\mathcal{E}$ have little to do with one another, in contrast with the well-understood cases of hyperbolic $2$- and $3$-manifolds. We also discuss growth of $b_1$ for congruence arithmetic lattices acting on $\mathbb{B}^2$ and $\mathbb{B}^3$. Along the way, we provide an explicit example of a lattice in $\mathrm{PU}(2, 1)$ admitting a homomorphism onto $\mathbb{Z}$ with finitely generated kernel. Moreover, we show that any cocompact arithmetic lattice $\Gamma \subset \mathrm{PU}(n, 1)$ of simplest type contains a finite index subgroup with this property.

We study a system of nonlinear singular integral equations with a sum-difference kernel on the positive half-line. Such a system occurs (in various representations) in many branches of mathematical physics and applied mathematics. In particular, one encounters a system of equations with a kernel representing a Gaussian distribution and with a power nonlinearity in the dynamic theory of p-adic open-closed strings; in the case when the nonlinearity has a certain exponential structure, such a system occurs in mathematical biology, namely, in the theory of the spatio-temporal distribution of an epidemic. We prove constructive theorems on the existence of nonnegative nontrivial continuous and bounded solutions. We study the uniqueness and the asymptotic behavior of constructed solutions at infinity. In conclusion, we give concrete applied examples of the mentioned equations that satisfy all assumptions of the proved theorems.

We introduce the Hadamard full propelinear codes that factorize as direct product of groups such that their associated group is \begin{document}$ C_{2t}\times C_2 $\end{document} . We study the rank, the dimension of the kernel, and the structure of these codes. For several specific parameters we establish some links from circulant Hadamard matrices and the nonexistence of the codes we study. We prove that the dimension of the kernel of these codes is bounded by \begin{document}$ 3 $\end{document} if the code is nonlinear. We also get an equivalence between circulant complex Hadamard matrix and a type of Hadamard full propelinear code, and we find a new example of circulant complex Hadamard matrix of order \begin{document}$ 16 $\end{document} .

We introduce the concept of a canonical subspace of C d [ z ] and among other results prove the following statements. An operator in a finite-dimensional vector space has no eigenvalues if and only if it is similar to the operator of multiplication by the independent variable on a canonical subspace of C d [ z ] . An operator in a finite-dimensional Pontryagin space is symmetric and has no eigenvalues if and only if it is isomorphic to the operator of multiplication by the independent variable in a canonical subspace of C d [ z ] with an inner product determined by a full matrix polynomial Nevanlinna kernel.

The primary objective of this present paper is to establish certain new weighted fractional Pólya–Szegö and Chebyshev type integral inequalities by employing the generalized weighted fractional integral involving another function Ψ in the kernel. The inequalities presented in this paper cover some new inequalities involving all other type weighted fractional integrals by applying certain conditions on $\omega (\theta )$ ω ( θ ) and $\Psi (\theta )$ Ψ ( θ ) . Also, the Pólya–Szegö and Chebyshev type integral inequalities for all other type fractional integrals, such as the Katugampola fractional integrals, generalized Riemann–Liouville fractional integral, conformable fractional integral, and Hadamard fractional integral, are the special cases of our main results with certain choices of $\omega (\theta )$ ω ( θ ) and $\Psi (\theta )$ Ψ ( θ ) . Additionally, examples of constructing bounded functions are also presented in the paper.

Inclusions resulting from the commutativity of elements and their commutators with trans\-vections in the language of residual and fixed submodules are found. The residual and fixed submodules of an element $\sigma $ of the complete linear group are defined as the image and the kernel of the element $\sigma -1$ and are denoted by $R(\sigma )$ and $P(\sigma )$, respectively. It is shown that for an arbitrary element $g$ of a complete linear group over a division ring whose characteristic is different from 2 and the transvection $\tau $ from the commutativity of the commutator $\left[g,\tau \right]$ with $g$ is followed by the inclusion of $R(\left[g,\tau \right])\subseteq P(\tau )\cap P(g)$. It is proved that the same inclusions occur over an arbitrary division ring if $g$ is a unipotent element, $\mathrm{dim}\mathrm{}(R\left(\tau \right)+R\left(g\right))\le 2$ and the commutator $\left[g,\tau \right]$ commutes with $\tau $ or if $g$ is a unipotent commutator of some element of the complete linear group and transvection $\ \tau $.

In this paper, we present the notion of perfect ideal of a seminearring S and prove that the kernel of a seminearring homomorphism is a perfect ideal. We show that the quotient structure S/I is isomorphic to the structure $$S_{T(I)}.$$ Finally, we prove isomorphism theorems in seminearrings by using tame condition.

In this paper, we describe the matrix representations of asymmetric truncated Toeplitz operators acting between two finite-dimensional model spaces$$K_1$$K1and$$K_2$$K2. The novelty of our approach is that here we consider matrix representations computed with respect to bases of different type in$$K_1$$K1and$$K_2$$K2(for example, kernel basis in$$K_1$$K1and conjugate kernel basis in$$K_2$$K2). We thus obtain new matrix characterizations which are simpler than the ones already known for asymmetric truncated Toeplitz operators.

Let $\Sigma$ be a surface with either boundary or marked points, equipped with an arbitrary framing. In this note we determine the action of the associated "framed mapping class group" on the homology of $\Sigma$ relative to its boundary (respectively marked points), describing the image as the kernel of a certain crossed homomorphism related to classical spin structures. Applying recent work of the authors, we use this to describe the monodromy action of the orbifold fundamental group of a stratum of abelian differentials on the relative periods., Comment: 18 pages. Final version, as appeared in B. Lond. Math. Soc

We study the Cantor–Bendixson rank of the space of subgroups for members of a general class of finitely generated self-replicating branch groups. In particular, we show for G either the Grigorchuk group or the Gupta–Sidki 3 group, the Cantor–Bendixson rank of Sub ( G ) is ω. For each natural number n, we additionally characterize the subgroups of rank n and give a description of subgroups in the perfect kernel.