Local Theorem Research Articles

Operators that are Weakly Coercive and a Compact Perturbation

ABSTRACT Let X, Y be two Banach spaces over K = ℝ \[\mathbb{K}=\mathbb{R}\] or ℂ \[\mathbb{C}\] , and let f := F+C be a weakly coercive operator from X onto Y, where F is a Fredholm proper operator, and C is a C 1-compact operator. Sufficient conditions are provided to assert that the perturbed operator f is a C 1-diffeomorphism. When one of these conditions does not hold and instead y is a regular value, the equation f(x) = y has at most finite number of solutions. As a consequence of the main result two corollaries are given. A second theorem studies the finite dimensional case. As an application, one example is given. The proof of our results is based on properties of Fredholm operators, as well as on local and global inverse mapping theorems.

Modal expansions of ririgs

Abstract In this paper, we introduce the variety of $I$-modal ririgs. We characterize the congruence lattice of its members by means of $I$-filters, and we provide a description of $I$-filter generation. We also provide an axiomatic presentation for the variety generated by chains of the subvariety of contractive $I$-modal ririgs. Finally, we introduce a Hilbert-style calculus for a logic with $I$-modal ririgs as an equivalent algebraic semantics and we prove that such a logic has the parametrized local deduction-detachment theorem.

The global structure theorem for finite groups with an abelian large p-subgroup

For a prime p, the Local Structure Theorem [15] studies finite groups G with the property that a Sylow p-subgroup S of G is contained in at least two maximal p-local subgroups. Under the additional assumptions that G contains a so called large p-subgroup Q≤S, and that composition factors of the normalizers of non-trivial p-subgroups are from the list of the known simple groups, [15] partially describes the p-local subgroups of G containing S, which are not contained in NG(Q). In the Global Structure Theorem, we extend the work of [15] and describe NG(Q) and, in almost all cases, the isomorphism type of the almost simple subgroup H generated by the p-local over-groups of S in G. Furthermore, for p=2, the isomorphism type of G is determined. In this paper, we provide a reduction framework for the proof of the Global Structure Theorem and also prove the Global Structure Theorem when Q is abelian.

A new type of Szász–Mirakjan operators based on q-integers

In this article, by using the notion of quantum calculus, we define a new type Szász–Mirakjan operators based on the q-integers. We derive a recurrence formula and calculate the moments Phi _{n,q}(t^{m};x) for m=0,1,2 and the central moments Phi _{n,q}((t-x)^{m};x) for m=1,2. We give estimation for the first and second-order central moments. We present a Korovkin type approximation theorem and give a local approximation theorem by using modulus of continuity. We obtain a local direct estimate for the new Szász–Mirakjan operators in terms of Lipschitz-type maximal function of order α. Finally, we prove a Korovkin type weighted approximation theorem.

Splash singularity for the free boundary incompressible viscous MHD

In this paper, we prove the existence of smooth initial data for the two-dimensional free boundary incompressible viscous magnetohydrodynamics (MHD) equations, for which the interface remains regular but collapses into a splash singularity (self-intersects in at least one point) in finite time. The existence of the splash singularities is guaranteed by a local existence theorem, in which we need suitable spaces for the modified magnetic field together with the modification of the velocity and pressure such that the modified initial velocity is zero, and a stability result which allows us to construct a class of initial velocities and domains for an arbitrary initial magnetic field. It turns out that the presence of the magnetic field does not prevent the viscous fluid from forming splash singularities for certain smooth initial data.

Generic Torelli and local Schottky theorems for Jacobian elliptic surfaces

Suppose that $f:X\to C$ is a general Jacobian elliptic surface over ${\mathbb {C}}$ of irregularity $q$ and positive geometric genus $h$. Assume that $10 h>12(q-1)$, that $h>0$ and let $\overline {\mathcal {E}\ell \ell }$ denote the stack of generalized elliptic curves. (1) The moduli stack $\mathcal {JE}$ of such surfaces is smooth at the point $X$ and its tangent space $T$ there is naturally a direct sum of lines $(v_a)_{a\in Z}$, where $Z\subset C$ is the ramification locus of the classifying morphism $\phi :C\to \overline {\mathcal {E}\ell \ell }$ that corresponds to $X\to C$. (2) For each $a\in Z$ the map $\overline {\nabla }_{v_a}:H^{2,0}(X)\to H^{1,1}_{\rm prim}(X)$ defined by the derivative $per_*$ of the period map $per$ is of rank one. Its image is a line ${\mathbb {C}}[\eta _a]$ and its kernel is $H^0(X,\Omega ^2_X(-E_a))$, where $E_a=f^{-1}(a)$. (3) The classes $[\eta _a]$ form an orthogonal basis of $H^{1,1}_{\rm prim}(X)$ and $[\eta _a]$ is represented by a meromorphic $2$-form $\eta _a$ in $H^0(X,\Omega ^2_X(2E_a))$ of the second kind. (4) We prove a local Schottky theorem; that is, we give a description of $per_*$ in terms of a certain additional structure on the vector bundles that are involved. Assume further that $8h>10(q-1)$ and that $h\ge q+3$. (5) Given the period point $per(X)$ of $X$ that classifies the Hodge structure on the primitive cohomology $H^2_{\rm prim}(X)$ and the image of $T$ under $per_*$ we recover $Z$ as a subset of ${\mathbb {P}}^{h-1}$ and then, by quadratic interpolation, the curve $C$. (6) We prove a generic Torelli theorem for these surfaces. Everything relies on the construction, via certain kinds of Schiffer variations of curves, of certain variations of $X$ for which $per_*$ can be calculated. (In an earlier version of this paper we used variations constructed by Fay. However, Schiffer variations are slightly more powerful.)

A "thoughtful" Local Friendliness no-go theorem: a prospective experiment with new assumptions to suit

A recent paper by two of us and co-workers \cite{CQD20}, based on an extended Wigner's friend scenario, demonstrated that certain empirical correlations predicted by quantum theory (QT) violate inequalities derived from a set of metaphysical assumptions we called "Local Friendliness" (LF). These assumptions are strictly weaker than those used for deriving Bell inequalities. Crucial to the theorem was the premise that a quantum system with reversible evolution could be an observer (colloquially, a "friend"). However, that paper was noncommittal on what would constitute an observer for the purpose of an experiment. Here, we present a new LF no-go theorem which takes seriously the idea that a system's having thoughts is a sufficient condition for it to be an observer. Our new derivation of the LF inequalities uses four metaphysical assumptions, three of which are thought-related, including one that is explicitly called "Friendliness". These four assumptions, in conjunction, allow one to derive LF inequalities for experiments involving the type of system that "Friendliness" refers to. In addition to these four metaphysical assumptions, this new no-go theorem requires two assumptions about what is technologically feasible: Human-Level Artificial Intelligence, and Universal Quantum Computing which is fast and large scale. The latter is often motivated by the belief that QT is universal, but this is not an assumption of the theorem. The intent of the new theorem is to give a clear goal for future experimentalists, and a clear motivation for trying to achieve that goal. We review various approaches to QT in light of our theorem. The popular stance that "quantum theory needs no interpretation" does not question any of our assumptions and so is ruled out. Finally, we quantitatively discuss how difficult the experiment we envisage would be, and briefly discuss milestones on the paths towards it.

A local rigidity theorem for minimal two-spheres in charged time-symmetric initial data set

A local rigidity theorem for minimal two-spheres in charged time-symmetric initial data set

On instability of a generic compressible two-fluid model in

This paper concerns the study of instability of a generic compressible two-fluid model in the whole space , where the capillary pressure is taken into account. For the case that the capillary pressure is a strictly decreasing function near the equilibrium, namely, , Evje et al (2016 Arch. Ration. Mech. Anal. 221 1285–316) established global stability of the constant equilibrium state for the three-dimensional Cauchy problem under some smallness assumptions. Recently, Wu et al (2022 arXiv:2204.10706) proved global stability of the constant equilibrium state for the case (corresponding to ). We investigate instability of the constant equilibrium state for the case that the capillary pressure is a strictly increasing function near the equilibrium, namely, . By employing the Hodge decomposition technique and making a detailed analysis of the Green’s function for the corresponding linearized system, we construct solutions of the linearized problem that grow exponentially in time in the Sobolev space Hk , thus leading to a global instability result for the linearized problem. Moreover, with the help of this global linear instability result together with a local existence theorem of classical solutions to the original nonlinear system, we can show instability of the nonlinear problem in the sense of Hadamard by carefully analysing the properties of the semigroup. Therefore, our results show that for the case , the constant equilibrium state of the two-fluid model is globally linearly unstable and locally nonlinearly unstable in the sense of Hadamard, which is different from the cases (Evje et al 2016 Arch. Ration. Mech. Anal. 221 1285–316) and (corresponding to ) (Wu et al 2022 arXiv:2204.10706) where the constant equilibrium state of the two-fluid model was proved to be globally nonlinearly stable.

Stochastic quantization of laser propagation models

This paper identifies certain interesting mathematical problems of stochastic quantization type in the modeling of Laser propagation through turbulent media. In some of the typical physical contexts, the problem reduces to stochastic Schrödinger equation with space–time white noise of Gaussian or Poisson or Lévy type. We identify their mathematical resolution via stochastic quantization. Nonlinear phenomena such as Kerr effect can be modeled by a stochastic nonlinear Schrödinger equation in the focusing case with space–time white noise. A treatment of stochastic transport equation, the Korteweg–De Vries equation as well as a number of other nonlinear wave equations with space–time white noise is also given. The main technique is the S-transform (we will actually use the closely related Hermite transform) which converts the stochastic partial differential equation (PDE) with space–time white noise to a deterministic PDE defined on the Hida–Kondratiev white noise distribution space. We then utilize the inverse S-transform/Hermite transform known as the characterization theorem combined with the infinite-dimensional implicit function theorem for analytic maps to establish local existence and uniqueness theorems for path-wise solutions of this class of problems. The particular focus of this paper on singular white noise distributions is motivated by practical situations where the refractive index fluctuations in the propagation medium in space and time are intense due to turbulence, ionospheric plasma turbulence, marine-layer fluctuations, etc. Since a large class of PDEs, that arise in nonlinear wave propagation, have polynomial-type nonlinearities, white noise distribution theory is an effective tool in studying these problems subject to different types of white noises.

The localisation theorem for the K-theory of stable ∞-categories

The localisation theorem for the <i>K</i>-theory of stable ∞-categories

On categories 𝒪 of quiver varieties overlying the bouquet graphs

We study representation theory of quantizations of Nakajima quiver varieties associated to bouquet quivers. We show that there are no finite dimensional representations of the quantizations A ¯ λ ( n , ℓ ) \overline {\mathcal {A}}_{\lambda }(n, \ell ) if both dim ⁡ V = n \operatorname {dim}V=n and the number of loops ℓ \ell are greater than 1 1 . We show that when n ≤ 3 n\leq 3 there is a Hamiltonian torus action with finitely many fixed points, provide the dimensions of Hom-spaces between standard objects in category O \mathcal {O} and compute the multiplicities of simples in standards for n = 2 n=2 in case of one-dimensional framing and generic one-parameter subgroups. We establish the abelian localization theorem and find the values of parameters, for which the quantizations have infinite homological dimension.

Higher Localization and Higher Branching Laws

Abstract For a connected reductive group $G$ and an affine smooth $G$-variety $X$ over the complex numbers, the localization functor takes $\mathfrak{g}$-modules to $D_{X}$-modules. We extend this construction to an equivariant and derived setting using the formalism of h-complexes due to Beilinson–Ginzburg, and show that the localizations of Harish-Chandra $(\mathfrak{g}, K)$-modules onto $X = H \backslash G$ have regular holonomic cohomologies when $H, K \subset G$ are both spherical reductive subgroups. The relative Lie algebra homologies and $\operatorname{Ext}$-branching spaces for $(\mathfrak{g}, K)$-modules are interpreted geometrically in terms of equivariant derived localizations. As direct consequences, we show that they are finite-dimensional under the same assumptions, and relate Euler–Poincaré characteristics to local index theorem; this recovers parts of the recent results of M. Kitagawa. Examples and discussions on the relation to Schwartz homologies are also included.

Normal forms of [formula omitted]-graded Q-manifolds

Following recent results of A.K. and V.S. on Z-graded manifolds, we give several local and global normal forms results for Q-structures on those, i.e. for differential graded manifolds. In particular, we explain in which sense their relevant structures are concentrated along the zero-locus of their curvatures, especially when the negative-part is of Koszul-Tate type. We also give a local splitting theorem.

Basic theorems in local class field theory and some further exploration

The relation between the abelian extension of a field and the topological groups of the field itself can be constructed using class field theory. In this piece of writing, the author will introduce the fundamental theorems of local class field theory by conducting a method of literature study. These theorems are the Reciprocity Law and the Existence Theorem, respectively. In addition to that, the author will discuss several unresolved issues in class field theory and provide examples of their applications in number theory. In class field theory, the results can be shown in two different ways. Both can be considered broad strokes. The first step is to demonstrate that the local case is true. By employing the methodology of cohomology and the theory of Lubin and Tate, one can demonstrate the Local Reciprocity Law and the Local Existence Theorem. The fundamental theorems in the global case are going to be demonstrated by utilizing the local results in conjunction with cohomology. Directly demonstrating the Gocal Reciprocity Law is another viable option.

Variational Approaches to a Discrete Elliptic Problem with a Weight

In this article, we are concerned with the existence of at least one, two and three distinct solutions for discrete boundary value problems driven by the Laplacian. The proof of the main result depends on variational methods. By using a consequence of the local minimum theorem due Bonanno we investigate the existence of at least one solution and two solutions for the problem with the weight. Furthermore, by using two critical point theorems, one due Averna and Bonanno, and another due Bonanno we explore the existence of two and three solutions for the problem. We also provide two examples in order to illustrate the main results.

Global existence of classical solutions for the gas flows near vacuum through ducts expanding with space and time

In this study, we examine the global existence of classical solutions for the gas flows near vacuum through ducts expanding with space and time. It is described by the initial-boundary value problem of compressible Euler equations together with a sufficiently small variable parameter, which can be viewed as a hyperbolic system of balance laws whose source is a non-dissipative term when Riemann invariants are applied. We prove a couple of global existence theorems of classical solutions under the suitable conditions of expanding ducts and the initial-boundary data whose C0 norms can be large. The analysis depends primarily on the local existence theorem and on uniform a priori estimates, which are obtained by giving the maximum principle and introducing new generalized Lax transformations. Furthermore, the limit behavior of expanding ducts at infinity and the long-time behavior of global classical solutions along all characteristic curves and vertical lines are also determined. Lastly, we explore the feasibility of the initial value problem for such expanding ducts.

Bubble-tree convergence and local diffeomorphism finiteness for gradient Ricci shrinkers

We prove bubble-tree convergence of sequences of gradient Ricci shrinkers with uniformly bounded entropy and uniform local energy bounds, refining the compactness theory of Haslhofer and Müller (Geom Funct Anal 21:1091–1116, 2011; Proc Am Math Soc 143(10):4433–4437, 2015). In particular, we show that no energy concentrates in neck regions, a result which implies a local energy identity for the sequence. Direct consequences of these results are an identity for the Euler characteristic and a local diffeomorphism finiteness theorem.

Extending the Domain with Application of Four-Step Nonlinear Scheme with Average Lipschitz Conditions

A novel local and semi-local convergence theorem for the four-step nonlinear scheme is presented. Earlier studies on local convergence were conducted without particular assumption on Lipschitz constant. In first part, the main local convergence theorems with a weak ϰ-average (assuming it as a positively integrable function and dropping the essential property of ND) are obtained. In comparison to previous research, in another part, we employ majorizing sequences that are more accurate in their precision along with the certain form of ϰ average Lipschitz criteria. A finer local and semi-local convergence criteria, boosting its utility, by relaxing the assumptions is derived. Applications in engineering to a variety of specific cases, such as object motion governed by a system of differential equations, are illustrated.

Representation Stability and Finite Orthogonal Groups

In this paper, we prove homological stability results about orthogonal groups over finite commutative rings where 2 is a unit. Inspired by Putman and Sam (2017), we construct a category OrI(R) and prove a local Noetherianity theorem for the category of OrI(R)-modules. This implies an asymptotic structure theorem for orthogonal groups. In addition, we show general homological stability theorems for orthogonal groups, with both untwisted and twisted coefficients.