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In this note we describe a method to calculate the action of a particular Fourier-Mukai transformation on a basis of brane charges on elliptically fibered Calabi-Yau threefolds with and without a section. The Fourier-Mukai kernel is the ideal sheaf of the relative diagonal and for fibrations that admit a section this is essentially the Poincar\'e sheaf. We find that in this case it induces an action of the modular group on the charges of 2-branes., Comment: 19 pages

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...

In this article, the authors characterize the Morrey spaces as well as their preduals via quadratic functions related to the Taylor remainder of the kernel of the Riesz potential. As applications, the authors obtain some strong capacitary inequalities, which are then used to study the regularity of the duality/weak solution to the fractional Laplace equation with measure data.

This work deals with the aggregation diffusion equation \[\partial_t \rho = \Delta_p\rho + \lambda div((K_a*\rho)\rho),\] where $K_a(x)=\frac{x}{|x|^a}$ is an attraction kernel and $\Delta_p$ is the so called $p$-Laplacian. We show that the domain $a < p(d+1)-2d$ is subcritical with respect to the competition between the aggregation and diffusion by proving that there is existence unconditionally with respect to the mass. In the critical case we show existence of solution in a small mass regime for an $L\ln L$ initial condition., Comment: 7 pages, 1 figure

It is well known that the Hormander smoothness condition sup y ≠ 0 ⁡ ∫ | x | ≥ 2 | y | | K ( x − y ) − K ( x ) | d x ∞ implies weak-type ( 1 , 1 ) estimates for associated L 2 -bounded Calderon–Zygmund operators. It has been an open question to know whether Hormander's condition also suffices to guarantee weak-type ( 1 , 1 , 1 / 2 ) estimates for bilinear Calderon–Zygmund operators that are bounded at one point. In this paper, we provide a negative answer to this question.

In this short note we extend some results obtained in \cite{Gazaki2015}. First, we prove that for an abelian variety $A$ with good ordinary reduction over a finite extension of $\mathbb{Q}_p$ with $p$ an odd prime, the Albanese kernel of $A$ is the direct sum of its maximal divisible subgroup and a torsion group. Second, for a semi-abelian variety $G$ over a perfect field $k$, we construct a decreasing integral filtration $\{F^r\}_{r\geq 0}$ of Suslin's singular homology group, $H_0^{sing}(G)$, such that the successive quotients are isomorphic to a certain Somekawa K-group.

We use our new type of bounded locally homeomorphic quasiregular mappings in the unit 3-ball to address long standing problems for such mappings, including the Vuorinen injectivity problem. The construction of such mappings comes from our construction of non-trivial compact 4-dimensional cobordisms $M$ with symmetric boundary components and whose interiors have complete 4-dimensional real hyperbolic structures. Such bounded locally homeomorphic quasiregular mappings are defined in the unit 3-ball $B^3\subset \mathbb{R}^3$ as mappings equivariant with the standard conformal action of uniform hyperbolic lattices $\Gamma\subset \operatorname{Isom} H^3$ in the unit 3-ball and with its discrete representation $G=\rho(\Gamma)\subset \operatorname{Isom} H^4 $. Here, $G$ is the fundamental group of our non-trivial hyperbolic 4-cobordism $M=(H^4\cup\Omega(G))/G,$ and the kernel of the homomorphism $\rho\!:\! Gamma\rightarrow G$ is a free group $F_3$ on three generators.

Kernel is an important topic in digraphs. A digraph such that every proper induced subdigraph has a kernel is said to be critical kernel imperfect (CKI, for short) if the digraph does not have a kernel. Galeana-Sanchez and Olsen characterized the CKI-digraphs for the following families of digraphs: asymmetric arc-locally in-/out-semicomplete digraphs, asymmetric 3-quasi-transitive digraphs and asymmetric 3-anti-quasi-transitive \(TT_3\)-free digraphs. In this paper, we shall completely characterize the above four classes CKI-digraphs without any restriction on arcs and \(TT_3\)-free subdigraphs.

The aim of this work is to show that the operator G, which has been introduced in Colombo et al. (Trans Am Math Soc 365:303–318, 2013) and whose kernel kerG coincides with the set of quaternionic slice regular functions, is a member of a family of operators with similar properties, such that all the members possess the respective versions of Stokes and Cauchy–type integral theorems. As direct consequences, these theorems are obtained for slice regular functions.

Congruences on an ample semigroup S are investigated. For any admissible congruence $$ \rho $$ on S, the minimum $$\sigma _{\rho }$$ (the maximum $$ \mu _{\rho }$$) admissible congruence on S whose restriction to E(S) is the trace of $$\rho $$ is obtained. An expression for a congruence which is contained in (contains) any admissible congruence of the same kernel is given. The concept of congruence pairs for S is introduced. It is shown that for any admissible congruence $$\rho $$ of S; ($${{\,\mathrm{tr}\,}}\rho , \ker \rho $$) is a congruence pair for S. Conversely, the congruence of S which contains (is contained in) any admissible congruence associated with a given congruence pair for S is formulated.

For an infinite set X, denote by Ω(X) the semigroup of all surjective mappings from X to X. We determine Green’s relations in Ω(X), show that the kernel (unique minimum ideal) of Ω(X) exists and determine its elements and cardinality. For a countably infinite set X, we describe the elements of Ω(X) for which the 𝒟-class and 𝒥-class coincide. We compare the results for Ω(X) with the corresponding results for other transformation semigroups on X.

Given an algebraic Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we canonically associate to it a Lie algebra $\mathfrak{g}_{\infty}$ defined over $\mathbb{C}_{\infty}$-the reduction of $\mathbb{C}$ mod infinitely large prime, and show that for a class of Lie algebras $\mathfrak{g}_{\infty}$ is an invariant of the derived category of $\mathfrak{g}$-modules. We give two applications of this construction. First, we show that the bounded derived category of $\mathfrak{g}$-modules determines algebra $\mathfrak{g}$ for a class of Lie algebras. Second, given a semi-simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we construct a canonical homomorphism from the group of automorphisms of the enveloping algebra $\mathfrak{U}\mathfrak{g}$ to the group of Lie algebra automorphisms of $\mathfrak{g}$, such that its kernel does not contain a nontrivial semi-simple automorphism. As a corollary we obtain that any finite subgroup of automorphisms of $\mathfrak{U}\mathfrak{g}$ isomorphic to a subgroup of Lie algebra automorphisms of $\mathfrak{g}.$, 12 pages, to appear in the IMRN

In this paper, we introduce the notion of involutive filters of pseudo-hoops, and we emphasize their role in the probability theory on these structures. Characterizations of involutive pseudo-hoops are given, and their properties are investigated. We give a characterization of involutive filters of a good pseudo-hoop, and we prove that in the case of good pseudo-hoops, the sets of fantastic and involutive filters are equal. One of the main results consists of proving that a normal filter F of a bounded pseudo-hoop A is involutive if and only if A / F is an involutive pseudo-hoop. It is also proved that any implicative filter of a bounded pseudo-hoop is involutive and that any Boolean filter of a bounded Wajsberg pseudo-hoop is involutive. The notions of state operators and state-morphism operators on pseudo-hoops are introduced, and the relationship between these operators is investigated. For a bounded Wajsberg pseudo-hoop, we prove that the kernel of any state operator is an involutive filter.

Let R be a commutative Noetherian ring, and let C be a semidualizing R-module. The notion of C-tilting R-modules is introduced as the relative setting of the notion of tilting R-modules with respect to C. Some properties of tilting and C-tilting modules and the relations between them are mentioned. It is shown that every finitely generated C-tilting R-module is C-projective. Finally, we investigate some kernel subcategories related to C-tilting modules.

We derive an identity relating the growth exponent of early-time OTOCs, the pre-exponential factor, and a third number called "branching time". The latter is defined within the dynamical mean-field framework, namely, in terms of the retarded kernel. This identity can be used to calculate stringy effects in the SYK and similar models, we also explicitly define "strings" in this context. As another application, we consider an SYK chain. If the coupling strength $\beta J$ is above a certain threshold and nonlinear (in the magnitude of OTOCs) effects are ignored, the exponent in the butterfly wavefront is exactly $2\pi/\beta$., Comment: 20 pages, 7 figures

For a given hereditary abelian category satisfying some finiteness conditions, in certain twisted cases it is shown that the modified Ringel-Hall algebra is isomorphic to the naive lattice algebra and there exists an epimorphism from the modified Ringel-Hall algebra to the lattice algebra. Furthermore, the kernel of this epimorphism is described explicitly. Finally, we show that the naive lattice algebra is invariant under the derived equivalences of hereditary abelian categories.

Let M be a 2n-dimensional smooth manifold with a symplectic pair which is a pair of closed 2-forms of constant ranks with complementary kernel foliations. Similar to Moser's stability theorem for symplectic forms, one desires to establish a stability theorem for symplectic pairs. Some sufficient and necessary conditions are obtained by Bande, Ghiggini and Kotschick. In this article, we consider a technical problem relating to the stability theorem. To complete the proof of the stability theorem for symplectic pairs, we verify the smoothness of the isotopy which is ignored in the literature. The Hodge theory for Riemannian foliation is crucial to our discussion.

Inspired by Okounkov's work [\emph{Selecta Mathematica}, 7(1):57--81, 2001] which relates KP hierarchy to determinant point process, we establish a relationship between BKP hierarchy and Pfaffian point process. We prove that the correlation function of the shifted Schur measures on strict partitions can be expressed as a Pfaffian of skew symmetric matrix kernel, whose elememts are certain vacuum expectations of neutral fermions. We further show that the matrix integrals solution of BKP hierarchy can also induce a certain Pfaffian point process., Comment: 17 pages

Z^d-extensions of probability-preserving dynamical systems are themselves dynamical systems preserving an infinite measure, and generalize random walks. Using the method of moments, we prove a generalized central limit theorem for additive functionals of the extension of integral zero, under spectral assumptions. As a corollary, we get the fact that Green-Kubo's formula is invariant under induction. This allows us to relate the hitting probability of sites with the symmetrized potential kernel, giving an alternative proof and generalizing a theorem of Spitzer. Finally, this relation is used to improve in turn the asumptions of the generalized central limit theorem. Applications to Lorentz gases in finite horizon and to the geodesic flow on abelian covers of compact manifolds of negative curvature are discussed., 56 pages, 3 figures

In this paper, we study a class of linear evolution equations of fractional order that are degenerate on the kernel of the operator under the sign of the derivative and on its relatively generalized eigenvectors. We prove that in the case considered, in contrast to the case of first-order degenerate equations and equations of fractional order with weak degeneration (i.e., degeneration only on the kernel of the operator under the sign of the derivative), the family of analytical in a sector operators does not vanish on relative generalized eigenspaces of the operator under the sign of the derivative, has a singularity at zero, and hence does not determine any solution of a strongly degenerate equation of fractional order. For the case of a strongly degenerate equation of integer order this fact does not hold, but the behavior of the family of resolving operators at zero cannot be examined by ordinary method.

We give new results of the phase transition of dilute colloidal solutions of rod-like molecules in dimension D ≥ 3 . For the low concentration of particles in a carrier fluid, we prove that the isotropic phase is the unique solution to the Doi–Onsager model with the general potential kernel. In addition, we present the regime of the bifurcation of nematic phases in the class of axially symmetric solutions. Our method is based on a generalization of the classical Leray–Schauder degree we developed for this problem.

We study inverse boundary problems for one dimensional linear integro-differential equation of the Gurtin-Pipkin type with the Dirichlet-to-Neumann map as the inverse data. Under natural conditions on the kernel of the integral operator, we give the explicit formula for the solution of the problem with the observation on the semiaxis t>0. For the observation on finite time interval, we prove the uniqueness result, which is similar to the local Borg-Marchenko theorem for the Schrodinger equation.

This paper is concerned with entire solutions of the monostable equation with nonlocal dispersal, i.e., \begin{document}$u_{t}=J*u-u+f(u)$\end{document} . Here the kernel \begin{document}$J$\end{document} is asymmetric. Unlike symmetric cases, this equation lacks symmetry between the nonincreasing and nondecreasing traveling wave solutions. We first give a relationship between the critical speeds \begin{document}$c^{*}$\end{document} and \begin{document}$\hat{c}^{*}$\end{document} , where \begin{document}$c^*$\end{document} and \begin{document}$\hat{c}^{*}$\end{document} are the minimal speeds of the nonincreasing and nondecreasing traveling wave solutions, respectively. Then we establish the existence and qualitative properties of entire solutions by combining two traveling wave solutions coming from both ends of real axis and some spatially independent solutions. Furthermore, when the kernel \begin{document}$J$\end{document} is symmetric, we prove that the entire solutions are 5-dimensional, 4-dimensional, and 3-dimensional manifolds, respectively.

In this paper, we study a class of integro-differential elliptic operators \begin{document} $L_{σ}$ \end{document} with kernel \begin{document} $k(y) = a(y)/|y|^{d+σ}$ \end{document} , where \begin{document} $d≥2, σ∈(0,2)$ \end{document} , and the positive function \begin{document} $a(y)$ \end{document} is homogenous and bounded. By using a purely analytic method, we construct the fundamental solution \begin{document} $Φ$ \end{document} of \begin{document} $L_{σ}$ \end{document} if \begin{document} $a(y)$ \end{document} satisfies a natural cancellation assumption and \begin{document} $|a(y)-1|$ \end{document} is small. Furthermore, we show that the fundamental solution \begin{document} $Φ$ \end{document} is \begin{document} $-α^{*}$ \end{document} homogeneous and Lipschitz continuous, where the constant \begin{document} $α^{*}∈(0,d)$ \end{document} . A Liouville-type theorem demonstrates that the fundamental solution \begin{document} $Φ$ \end{document} is the unique nontrivial solution of \begin{document} $L_{σ}u = 0$ \end{document} in \begin{document} $\mathbb{R}^{d}\setminus\{0\}$ \end{document} that is bounded from below.

It is known that the equation $x'(t)=Ax(t)+f(t)$, where $A$ is a bounded linear operator, has a unique bounded solution $x$ for any bounded continuous free term~$f$ if and only if the spectrum of the coefficient $A$ does not intersect the imaginary axis. The solution can be represented in the form \begin{equation*} x(t)=\int_{-\infty}^{\infty}\mathcal G(s)f(t-s)\,ds. \end{equation*} The kernel $\mathcal G$ is called Green's function. In this paper, the case when $A$ admits a representation by a block triangular operator matrix is considered. It is shown that the blocks of $\mathcal G$ are sums of special convolutions of Green's functions of diagonal blocks of $A$.

We are concerned with Holder regularity estimates for weak solutions \begin{document}$u$\end{document} to nonlocal Schrodinger equations subject to exterior Dirichlet conditions in an open set \begin{document}$\Omega\subset \mathbb{R}^N$\end{document} . The class of nonlocal operators considered here are defined, via Dirichlet forms, by symmetric kernels \begin{document}$K(x, y)$\end{document} bounded from above and below by \begin{document}$|x-y|^{-N-2s}$\end{document} , with \begin{document}$s\in (0, 1)$\end{document} . The entries in the equations are in some Morrey spaces and the domain \begin{document}$\Omega$\end{document} satisfies some mild regularity assumptions. In the particular case of the fractional Laplacian, our results are new. When \begin{document}$K$\end{document} defines a nonlocal operator with sufficiently regular coefficients, we obtain Holder estimates, up to the boundary of \begin{document}$ \Omega$\end{document} , for \begin{document}$u$\end{document} and the ratio \begin{document}$u/d^s$\end{document} , with \begin{document}$d(x) = \text{dist}(x, \mathbb{R}^N\setminus\Omega)$\end{document} . If the kernel \begin{document}$K$\end{document} defines a nonlocal operator with Holder continuous coefficients and the entries are Holder continuous, we obtain interior \begin{document}$C^{2s+\beta}$\end{document} regularity estimates of the weak solutions \begin{document}$u$\end{document} . Our argument is based on blow-up analysis and compact Sobolev embedding.

In this paper, we extend the Atiyah--Guillemin--Sternberg convexity theorem and Delzant's classification of symplectic toric manifolds to presymplectic manifolds. We also define and study the Morita equivalence of presymplectic toric manifolds and of their corresponding framed momentum polytopes, which may be rational or non-rational. Toric orbifolds, quasifolds and non-commutative toric varieties may be viewed as the quotient of our presymplectic toric manifolds by the kernel isotropy foliation of the presymplectic form.

We discuss the kernel of the localization map from étale motivic cohomology of a variety over a number field to étale motivic cohomology of the base change to its completions. This generalizes the Hasse principle for the Brauer group, and is related to Tate–Shafarevich groups of abelian varieties.

In this article, benefiting from the nabla h − fractional functions and nabla h − Taylor polynomials, some properties of the nabla h − discrete version of Mittag-Leffler ( h − ML) function are studied. The monotonicity of the nabla h − fractional difference operator with h − ML kernel (Atangana–Baleanu fractional differences) is discussed. As an application, the Mean Value Theorem (MVT) on h Z is proved.

The commutator subgroup of a free group is the subgroup consisting of elements contained in the kernel of every homomorphism from the free group to the integers. We give a similar characterization of the second derived subgroup of a free group. Specifically, we show that the second derived subgroup of a free group is equal to the intersection of the kernels of all homomorphisms into a solvable deficiency 1 group.

In this paper we investigate to what extent the results of Z. Wang and D. Daigle on “nice derivations” of the polynomial ring k [ X , Y , Z ] over a field k of characteristic zero extend to the polynomial ring R [ X , Y , Z ] over a PID R , containing the field of rational numbers. One of our results shows that the kernel of a nice derivation on k [ X 1 , X 2 , X 3 , X 4 ] of rank at most three is a polynomial ring over k .

We consider the Fisher–KPP equation with a non-local interaction term. In [16] , Hamel and Ryzhik showed that in solutions of this equation, the front location at a large time t is 2 t + o ( t ) . We study the asymptotics of the second order term in the front location. If the interaction kernel ϕ ( x ) decays sufficiently fast as x → ∞ then this term is given by − 3 2 2 log ⁡ t + o ( log ⁡ t ) , which is the same correction as found by Bramson in [7] for the local Fisher–KPP equation. However, if ϕ has a heavier tail then the second order term is − t β + o ( 1 ) , where β ∈ ( 0 , 1 ) depends on the tail of ϕ. The proofs are probabilistic, using a Feynman–Kac formula. Since solutions of the non-local Fisher–KPP equation do not obey the maximum principle, the proofs differ from those in [7] , although some of the ideas used are similar.

WOS:000452702300001 In this manuscript, we prove new aspects for several Opial-type integral inequalities for the left and right Caputo-Fabrizio operators with nonsingular kernel. For this purpose we use the inequalities obtained by Andri et al. (Integral Transforms Spec. Funct. 25(4):324-335, 2014), which is the generalization of an inequality of Agarwal and Pang (Opial Inequalities with Applications in Differential and Difference Equations, 1995). Besides, examples are presented to validate the reported results.

We study the global existence and the large time behavior of solutions for the complex-valued quadratic heat equation: $$\partial _t z=\mathcal {L}\,z+z^2,\,t>0,\,x\in \mathbb {R}^{N},$$ with initial data $$z_0=u_0+i\,v_0$$ . $$\mathcal {L}$$ is a linear operator with $$e^{t\mathcal {L}}$$ its semigroup having a generalized heat kernel G satisfying in particular $$G(t,x)= t^{-\frac{N}{d}} G(1,xt^{-1/d}),\,d>0,\, t>0$$ and $$x\in \mathbb {R}^N.$$ For $$\sigma \ge 1,\, 2\rho -\sigma \ge 1$$ , $$\frac{N}{{\text {d}}\sigma }>1,$$ $$\frac{N}{{\text {d}}\rho }>1$$ and if $$u_{0}(x)\sim c|x|^{-{\text {d}}\sigma }$$ and $$v_{0}(x)\sim c|x|^{-{\text {d}}\rho },$$ as $$|x|\rightarrow \infty $$ (|c| is small) we prove the global existence and we establish the existence of four different asymptotically self-similar behaviors. Our results apply for the fractional and higher order complex-valued quadratic heat equation and are new even in these cases.

Let $$H = {\mathcal H}\oplus {\mathcal K}$$ be the direct sum of two Hilbert spaces. In this paper we characterise the semi-projections (defined in the paper) and projections with a given kernel and a given range that can be described by a two by two matrix or block of relations determined by the decompositions of $${\mathcal H}= {\mathcal H}_{1} \oplus {\mathcal H}_{2}$$ and of $${\mathcal K}= {\mathcal K}_{1} \oplus {\mathcal K}_{2}$$ . This generalises the Stone - de Snoo (Oral communication to the author, 1992; J Indian Math Soc 15: 155–192, 1952) formula for the orthogonal projection on the graph of a closed linear relation, and extends the results of Mezroui (Trans AMS 352: 2789–2800, 1999) on the same subject. This requires some new results concerning blocks of linear relations as studied in (Adv Oper Theory 5: 1193–1228, 2020). Some applications are given on the product of two relations including one contained in (Complex Anal Oper Theory 6: 613–624, 2012).

The Baumslag-Solitar group $BS(2,3)$, is a so-called non-Hopfian group, meaning that it has an epimorphism $\phi$ onto itself, that is not injective. In particular this is equivalent to saying that $BS(2,3)$ has a non-trivial quotient that is isomorphic to itself. As a consequence the Cayley graph of $BS(2,3)$ has a quotient that is isomorphic to itself up to change of generators. We describe this quotient on the graph-level and take a closer look at the most common epimorphism $\phi$. We show its kernel is a free group of infinite rank with an explicit set of generators. Finally we show how $\phi$ appears as a morphism on fundamental groups induced by some continuous map. This point of view was communicated to the author by Gilbert Levitt., Comment: 12 pages, 13 figures, comments welcome

Funding: the research of the last two authors is supported by the Australian Research Council Discovery Project DP200101951. This work was supported by EPSRC grant no EP/R014604/1. In addition, the second author was supported by a Simons Fellowship. We examine the subgroup D(G) of a transitive permutation group G which is generated by the derangements in G. Our main results bound the index of this subgroup: we conjecture that, if G has degree n and is not a Frobenius group, then |G:D(G)|≤ √n-1; we prove this except when G is a primitive affine group. For affine groups, we translate our conjecture into an equivalent form regarding |H:R(H)|, where H is a linear group on a finite vector space and R(H) is the subgroup of H generated by elements having eigenvalue 1. If G is a Frobenius group, then D(G) is the Frobenius kernel, and so G/D(G) is isomorphic to a Frobenius complement. We give some examples where D(G) ≠ G, and examine the group-theoretic structure of G/D(G); in particular, we construct groups G in which G/D(G) is not a Frobenius complement. Postprint

Two-dimensional conformal field theories with Virasoro symmetry generically contain a Schwarzian sector. This sector is related to the near-horizon region of the near-extremal BTZ black hole in the holographic dual. In this work we generalize this picture to CFTs with higher spin conserved currents. It is shown that the partition function in the near-extremal limit agrees with that of BF higher spin gravity in AdS$_2$ which is described by a generalized Schwarzian theory. We also provide a spectral decomposition of Schwarzian partition functions via the $\mathcal{W}_N$ fusion kernel and consider supersymmetric generalizations., Comment: v2: 22 pages, approximates published version

This manuscript concerns two definite integrals that could be connected to the Bose-Einstein and the Fermi-Dirac functions in the integrands, separately, with numerators slightly modified with a difference in two expressions that contain the Fourier kernel multiplied by a polynomial and its complex conjugate. In this work, we use our contour integral method to derive these definite integrals, which are given by ∫0∞ie−imx(log(a)−ix)k−eimx(log(a)+ix)k2eαx−1dx and ∫0∞ie−imx(log(a)−ix)k−eimx(log(a)+ix)k2eαx+1dx in terms of the Lerch function. We use these two definite integrals to derive formulae by Erdéyli and Watson. We derive special cases of these integrals in terms of special functions not found in current literature. Special functions have the property of analytic continuation, which widens the range of computation of the variables involved.

We consider a compact CR manifold with a transversal CR locally free circle action endowed with an S1-equivariant positive CR line bundle. We prove that a certain weighted Fourier–Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion. As a consequence, we establish an equivariant Kodaira embedding theorem.

This article aims to extend the wavelet transform to quaternion algebra using the kernel of the two-sided quaternion Fourier transform. We study some fundamental properties of this extension such as scaling, translation, rotation, and Parseval’s identity, then we derive the associated Heisenberg–Pauli–Weyl uncertainty principle UP. Finally, using the quaternion Fourier representation of the CQWT we generalize the logarithmic UP to the CQWT domain.

Linear recurrent sequences are those whose elements are defined as linear combinations of preceding elements, and finding recurrence relations is a fundamental problem in computer algebra. In this paper, we focus on sequences whose elements are vectors over the ring $\mathbb{A} = \mathbb{K}[x]/(x^d)$ of truncated polynomials. Finding the ideal of their recurrence relations has applications such as the computation of minimal polynomials and determinants of sparse matrices over $\mathbb{A}$. We present three methods for finding this ideal: a Berlekamp-Massey-like approach due to Kurakin, one which computes the kernel of some block-Hankel matrix over $\mathbb{A}$ via a minimal approximant basis, and one based on bivariate Pad\'e approximation. We propose complexity improvements for the first two methods, respectively by avoiding the computation of redundant relations and by exploiting the Hankel structure to compress the approximation problem. Then we confirm these improvements empirically through a C++ implementation, and we discuss the above-mentioned applications., Comment: 8 pages, ISSAC 2021

We discuss various aspects of the positive kernel method of quantization of the one-parameter groups $$\tau _t \in \text{ Aut }(P,\vartheta )$$ τ t ∈ Aut ( P , ϑ ) of automorphisms of a G-principal bundle $$P(G,\pi ,M)$$ P ( G , π , M ) with a fixed connection form $$\vartheta $$ ϑ on its total space P. We show that the generator $${\hat{F}}$$ F ^ of the unitary flow $$U_t = e^{it {\hat{F}}}$$ U t = e i t F ^ being the quantization of $$\tau _t $$ τ t is realized by a generalized Kirillov–Kostant–Souriau operator whose domain consists of sections of some vector bundle over M, which are defined by a suitable positive kernel. This method of quantization applied to the case when $$G=\hbox {GL}(N,{\mathbb {C}})$$ G = GL ( N , C ) and M is a non-compact Riemann surface leads to quantization of the arbitrary holomorphic flow $$\tau _t^{\mathrm{hol}} \in \text{ Aut }(P,\vartheta )$$ τ t hol ∈ Aut ( P , ϑ ) . For the above case, we present the integral decompositions of the positive kernels on $$P\times P$$ P × P invariant with respect to the flows $$\tau _t^{\mathrm{hol}}$$ τ t hol in terms of the spectral measure of $${\hat{F}}$$ F ^ . These decompositions generalize the ones given by Bochner’s Theorem for the positive kernels on $${\mathbb {C}} \times {\mathbb {C}}$$ C × C invariant with respect to the one-parameter groups of translations of complex plane.

The goal of this present paper is to study some new inequalities for a class of differentiable functions connected with Chebyshev’s functionals by utilizing a fractional generalized weighted fractional integral involving another function$\mathcal{G}$Gin the kernel. Also, we present weighted fractional integral inequalities for the weighted and extended Chebyshev’s functionals. One can easily investigate some new inequalities involving all other type weighted fractional integrals associated with Chebyshev’s functionals with certain choices of$\omega (\theta )$ω(θ)and$\mathcal{G}(\theta )$G(θ)as discussed in the literature. Furthermore, the obtained weighted fractional integral inequalities will cover the inequalities for all other type fractional integrals such as Katugampola fractional integrals, generalized Riemann–Liouville fractional integrals, conformable fractional integrals and Hadamard fractional integrals associated with Chebyshev’s functionals with certain choices of$\omega (\theta )$ω(θ)and$\mathcal{G}(\theta )$G(θ).

Fractional analysis has evolved considerably over the last decades and has become popular in many technical and scientific fields. Many integral operators which ables us to integrate from fractional orders has been generated. Each of them provides different properties such as semigroup property, singularity problems etc. In this paper, firstly, we obtained a new kernel, then some new integral inequalities which are valid for integrals of fractional orders by using Riemann-Liouville fractional integral. To do this, we used some well-known inequalities such as Hölder's inequality or power mean inequality. Our results generalize some inequalities exist in the literature.

A conformal mapping function that maps a multiply connected region onto a unit disk is known as the Ahlfors map. It reduces to the Riemann map for a simply connected region. The Ahlfors map can be expressed in terms of the Szego kernel. The Szego kernel is a unique solution to a second-kind Fredholm integral equation. The Ahlfors map and the Szego kernel have the same zeros. There is no known explicit formula for the zeros of the Ahlfors map for doubly connected regions except for the annulus region. For general doubly connected regions, the zeros of the Ahlfors map must be computed numerically. This paper shows how to determine the zeros of the Ahlfors map numerically for any doubly connected regions with smooth boundaries. The numerical examples and comparisons are also presented.

We investigate configuration-space integrals over punctured Riemann spheres from the viewpoint of the motivic Galois coaction and double-copy structures generalizing the Kawai-Lewellen-Tye (KLT) relations in string theory. For this purpose, explicit bases of twisted cycles and cocycles are worked out whose orthonormality simplifies the coaction. We present methods to efficiently perform and organize the expansions of configuration-space integrals in the inverse string tension $\alpha'$ or the dimensional-regularization parameter $\epsilon$ of Feynman integrals. Generating-function techniques open up a new perspective on the coaction of multiple polylogarithms in any number of variables and analytic continuations in the unintegrated punctures. We present a compact recursion for a generalized KLT kernel and discuss its origin from intersection numbers of Stasheff polytopes and its implications for correlation functions of two-dimensional conformal field theories. We find a non-trivial example of correlation functions in $(\mathfrak{p},2)$ minimal models, which can be normalized to become uniformly transcendental in the $\mathfrak{p} \to \infty$ limit., Comment: 100 pages, 7 figures, journal version

In the context of inhomogeneous one-dimensional finite systems, recent numerical advances [Phys. Rev. B 103, 125155 (2021)] allow us to compute the exact coupling-constant dependent exchange-correlation kernel ${f}_{\text{xc}}^{\ensuremath{\lambda}}(x,{x}^{\ensuremath{'}},\ensuremath{\omega})$ within linear response time-dependent density-functional theory. This permits an improved understanding of ground-state total energies derived from the adiabatic-connection fluctuation-dissipation theorem (ACFDT). We consider both one-shot and self-consistent ACFDT calculations, and demonstrate that chemical accuracy is reliably preserved when the frequency dependence in the exact functional ${f}_{\text{xc}}[n](\ensuremath{\omega}=0)$ is neglected. This performance is understood on the grounds that the exact ${f}_{\text{xc}}[n]$ varies slowly over the most relevant $\ensuremath{\omega}$ range (but not in general), and hence the spatial structure in ${f}_{\text{xc}}[n](\ensuremath{\omega}=0)$ is able to largely remedy the principal issue in the present context: self-interaction (examined from the perspective of the exchange-correlation hole). Moreover, we find that the implicit orbitals contained within a self-consistent ACFDT calculation utilizing the adiabatic exact kernel ${f}_{\text{xc}}[n](\ensuremath{\omega}=0)$ are remarkably similar to the exact Kohn-Sham orbitals, thus further establishing that the majority of the physics required to capture the ground-state total energy resides in the spatial dependence of ${f}_{\text{xc}}[n]$ at $\ensuremath{\omega}=0$.

In this article we expand the notion of (δ, γ) - fuzzy kernel of a (δ, γ) - fuzzy homomorphism on rings and also we show that it is a (δ, γ) - fuzzy ideal of the domain ring. We can also prove that any (δ, γ) - fuzzy ideal of a ring is a (δ, γ) - fuzzy kernel of some (δ, γ) – fuzzy epimorphism, namely the canonical (δ, γ) – fuzzy epimorphism. End of this article we formulate and proven the fundamental theorem of (δ, γ) - fuzzy homomorphism and the first, the second and the third (δ, γ) - fuzzy isomorphism theorem on rings.