Proof Of Theorem Research Articles

The K-functional and reiteration theorems for Left and Right spaces, Part II

We consider the K-interpolation methods involving slowly varying functions. Let A‾θ,⁎L and A‾θ,⁎R (0≤θ≤1) be the so-called L or R limiting interpolation spaces which arise naturally in reiteration formulae for the limiting cases. We characterize the interpolation spaces (X,Y)θ,r,a (0≤θ≤1), where X=A‾θ0,⁎L or X=A‾θ0,⁎R and Y=A‾θ1,⁎L or Y=A‾θ1,⁎R, provided that 0≤θ0<θ1≤1 and at least one of the parameters θ0 or θ1 has the limiting value θ0=0 or θ1=1. This supplements the first part of the paper, where one of the operands is standard interpolation spaces involving slowly varying functions. The proofs of most reiteration theorems are based on Holmstedt-type formulae. Applications to grand and small Lorentz spaces are given.

Bogomol’nyi-like equations in gravity theories

Using the Bogomol’nyi–Prasad–Sommerfield Lagrangian method, we show that gravity theory coupled to matter in various dimensions may possess Bogomol’nyi-like equations, which are first-order differential equations, satisfying the Einstein equations and the Euler–Lagrange equations of classical fields (U(1) gauge and scalar fields). In particular we consider static and spherically symmetric solutions by taking proper ansatzes and then we find an effective Lagrangian density that can reproduce the Einstein equations and the Euler–Lagrange equations of the classical fields. We consider the BPS Lagrangian density to be linear function of first-order derivative of all the fields. From these two Lagrangian desities we are able to obtain the Bogomol’nyi-like equations whose some of solutions are well-known such as Schwarzschild, Reissner–Nordström, Tangherlini black holes, and the recent black holes with scalar hair in three dimensions (Phys. Rev. D 107, 124047). Using these Bogomol’nyi-like equations, we are also able to find new solutions for scalar hair black holes in three and four dimensional spacetime. Furthermore we show that the Bogomol’nyi–Prasad–Sommerfield Lagrangian method can provide a simple alternative proof of black holes uniqueness theorems in any dimension.

Elementary proof of Funahashi's theorem

Funahashi established that the space of two-layer feedforward neural networks is dense in the space of all continuous functions defined over compact sets in $n$-dimensional Euclidean space. The purpose of this short survey is to reexamine the proof of Theorem 1 in Funahashi \cite{Funahashi}. The Tietze extension theorem, whose proof is contained in the appendix, will be used. This paper is based on harmonic analysis, real analysis, and Fourier analysis. However, the audience in this paper is supposed to be researchers who do not specialize in these fields of mathematics. Some fundamental facts that are used in this paper without proofs will be collected after we present some notation in this paper.

Fermatean Neutrosophic INK-algebras

This paper introduces the concept of the direct product of sets that involve Fermatean neutrosophic elements in structures called INK-algebras. It defines terms like the direct product of Fermatean neutrosophic INK-ideals in INK-algebras and Fermatean neutrosophic sets (FNSs), Fermatean neutrosophic INK-ideals (FNINK-Is), and Fermatean neutrosophic closed INK-ideals (FNCINK-Is). The proof of theorems illustrating the relationships between these ideas is included in the paper. It also defines the INK-sub algebra embedded in an INK-algebra and gives a theorem elucidating the connection between the direct product of Fermatean neutrosophic INK-ideals and the images of these sub-algebras. In essence, the paper investigates and establishes connections between different mathematical ideas concerning INK-algebras and FNSs.

A note on 𝐶^{1,𝛼}-smooth approximation of Lipschitz functions

We show that on super-reflexive spaces a Moreau-Yosida type of regularisation by infimal convolution together with a known insertion-type theorem (a variant of Ilmanen’s lemma) easily give an approximation of a Lipschitz function by a C 1 , α C^{1,\alpha } -smooth Lipschitz function with the same Lipschitz constant. This is a generalisation of the well-known theorem of J.-M. Lasry and P.-L. Lions from Hilbert spaces. It also gives a new self-contained and probably simpler proof of the Lasry-Lions theorem.

Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation

Abstract This paper is devoted to studying the pointwise convergence problem and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation. Firstly, we present an alternative proof of Theorem 1.5 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936] and Theorem 1.8 of Linares and Ramos [The Cauchy problem for the L 2 L^{2} -critical generalized Zakharov–Kuznetsov equation in dimension 3, Comm. Partial Differential Equations 46 (2021), 9, 1601–1627]. Secondly, we give an alternative proof of Theorem 1.1 of Ribaud and Vento [A note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations, C. R. Math. Acad. Sci. Paris 350 (2012), 9–10, 499–503] and present the nonlinear smoothing and uniform convergence of two-dimensional generalized Zakharov–Kuznetsov equation. Thirdly, we give an alternative proof of Theorem 1.4 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936]. Finally, we study the nonlinear smoothing and uniform convergence of 𝑛-dimensional generalized Zakharov–Kuznetsov equation with n ≥ 3 n\geq 3 .

A variational proof of a disentanglement theorem for multilinear norm inequalities

The basic disentanglement theorem established by the present authors states that estimates on a weighted geometric mean over (convex) families of functions can be disentangled into quantitatively linked estimates on each family separately. On the one hand, the theorem gives a uniform approach to classical results including Maurey's factorisation theorem and Lozanovskiĭ's factorisation theorem, and, on the other hand, it underpins the duality theory for multilinear norm inequalities developed in our previous two papers.In this paper we give a simple proof of this basic disentanglement theorem. Whereas the approach of our previous paper was rather involved – it relied on the use of minimax theory together with weak*-compactness arguments in the space of finitely additive measures, and an application of the Yosida–Hewitt theory of such measures – the alternate approach of this paper is rather straightforward: it instead depends upon elementary perturbation and compactness arguments.

An analytic proof of the stable reduction theorem

AbstractThe stable reduction theorem says that a family of curves of genus $$g\ge 2$$ g ≥ 2 over a punctured curve can be uniquely completed (after possible base change) by inserting certain stable curves at the punctures. We give a new this result for curves defined over $${\mathbb {C}}$$ C , using the Kähler–Einstein metrics on the fibers to obtain the limiting stable curves at the punctures.

From port-based teleportation to Frobenius reciprocity theorem: partially reduced irreducible representations and their applications

In this paper, we present the connection of two concepts as induced representation and partially reduced irreducible representations (PRIR) appear in the context of port-based teleportation protocols. Namely, for a given finite group G with arbitrary subgroup H, we consider a particular case of matrix irreducible representations, whose restriction to the subgroup H, as a matrix representation of H, is completely reduced to diagonal block form with an irreducible representation of H in the blocks. The basic properties of such representations are given. Then as an application of this concept, we show that the spectrum of the port-based teleportation operator acting on n systems is connected in a very simple way with the spectrum of the corresponding Jucys–Murphy operator for the symmetric group S(n-1)⊂S(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S(n-1)\\subset S(n)$$\\end{document}. This shows on the technical level relation between teleporation and one of the basic objects from the point of view of the representation theory of the symmetric group. This shows a deep connection between the central object describing properties of deterministic PBT schemes and objects appearing naturally in the abstract representation theory of the symmetric group. In particular, we present a new expression for the eigenvalues of the Jucys–Murphy operators based on the irreducible characters of the symmetric group. As an additional but not trivial result, we give also purely matrix proof of the Frobenius reciprocity theorem for characters with explicit construction of the unitary matrix that realizes the reduction in the natural basis of induced representation to the reduced one.

A synthetic proof of the Morley trisector theorem using congruent and similar triangles

A synthetic proof of the Morley trisector theorem using congruent and similar triangles

Group invariant variational principles

In this paper we introduce a group invariant version of the well-known Ekeland variational principle. To achieve this, we define the concept of convexity with respect to a group and establish a version of the theorem within this framework. Additionally, we present several consequences of the group invariant Ekeland variational principle, including Palais-Smale minimizing sequences, the Brønsted-Rockafellar theorem, and a characterization of the linear and continuous group invariant functionals space. Moreover, we provide an alternative proof of the Bishop-Phelps theorem and proofs for the group-invariant Hahn-Banach separating theorems. Finally, we discuss some implications and applications of these results.

Corrigendum: Model theory of fields with virtually free group actions

AbstractThere is an irreparable error in the proof of Theorem 3.26 in the above‐mentioned paper and we withdraw the claim of having proved that theorem. In fact, that theorem is false in a very strong sense.

Optimal implementation of quantum gates with two controls

We give a detailed proof of a well-known theorem in quantum computing. The theorem characterizes the number of two-qubit gates that is necessary for implementing three-qubit quantum gates with two controls. For example, the theorem implies that five 2-qubit gates are necessary for implementing the Toffoli gate. No detailed proof was available earlier.

Correction to: Generalization of Hamiltonian mechanics to a three-dimensional phase space

Abstract In a recent paper [N. Sato, Prog. Theor. Exp. Phys. 2021, 6, 063A01 (2021)] we introduced a generalization of Hamiltonian mechanics to three-dimensional phase spaces in terms of closed 3-forms. This correction addresses an error in the proof of theorem 3, which concerns the existence of a coordinate change transforming a closed 3-form into a constant form. Indeed, invertibility of a 3-form is not sufficient to ensure the existence of a solution Xt to eq. (77) when n &amp;gt; 3. The theorem can be corrected by restricting the class of 3-forms to those that are relevant to generalized Hamiltonian mechanics. Although the new theorem requires a stronger hypothesis, the formulation of dynamical systems with 2 invariants in terms of closed 3-forms, which is the key contribution of the paper, holds.

Solvability of an Inverse Problem for an Elliptic-Type Equation

In this study, we consider an inverse problem of determining an unknown source function in the right-hand side of an elliptic equation which is ill-posed in the Hadamard sense. To investigate the solvability of the problem, we reduce it to a Dirichlet problem for a third-order partial differential equation with homogeneous boundary condition. Since the problem is linear, the proof of the uniqueness theorem is based on the Fredholm Alternative Theorem. We prove the existence of the solution to the problem by using the Galerkin method.

A short proof of Tomita's theorem

Tomita-Takesaki theory associates a positive operator called the “modular operator” with a von Neumann algebra and a cyclic-separating vector. Tomita's theorem says that the unitary flow generated by the modular operator leaves the algebra invariant. I give a new, short proof of this theorem which only uses the analytic structure of unitary flows, and which avoids operator-valued Fourier transforms (as in van Daele's proof) and operator-valued Mellin transforms (as in Zsidó's and Woronowicz's proofs). The proof is similar to one given by Bratteli and Robinson in the special case that the modular operator is bounded.

Corrigendum to “The discriminant of compositum of algebraic number fields”

We point out that there is an error in the proof of Theorem 1.1 in [The discriminant of compositum of algebraic number fields, Int. J. Number Theory 15 (2019) 353–360]. We also prove that the result of this theorem holds with an additional hypothesis. However, it is an open problem whether the result of the theorem is true in general or not.

Cohomology and geometry of Deligne–Lusztig varieties for GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document}

We give a description of the cohomology groups of the structure sheaf on smooth compactifications X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document} of Deligne–Lusztig varieties X(w) for GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document}, for all elements w in the Weyl group. As a consequence, we obtain the modpm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{mod}\\ p^m$$\\end{document} and integral p-adic étale cohomology of X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document}. Moreover, using our result for X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document} and a spectral sequence associated to a stratification of X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document}, we deduce the modpm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{mod}\\ p^m$$\\end{document} and integral p-adic étale cohomology with compact support of X(w). In our proof of the main theorem, in addition to considering the Demazure–Hansen smooth compactifications of X(w), we show that a similar class of constructions provide smooth compactifications of X(w) in the case of GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document}. Furthermore, we show in the appendix that the Zariski closure of X(w), for any connected reductive group G and any w, has pseudo-rational singularities.

Simple proof of the global inverse function theorem via the Hopf–Rinow theorem

Simple proof of the global inverse function theorem via the Hopf–Rinow theorem

The Ptolemy-Euler Theorem via Reflection

Summary Although there are several generalizations of the Ptolemy-Euler theorem in plane geometry, its higher dimensional cases are still interesting enough to deserve attention. In this note, we give a new proof of the Ptolemy-Euler theorem in R n , via a modified matrix version (reflection transformation) of a classical algebraic identity.