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    • The first result of this paper is

    2023-09-14

    The first result of this paper is parallel to the positivity result given in [16]: Here the definition of approximate ACHE metric is the same as in [18]. See Definition 1 for details. The second result of this paper is parallel to the (-)-Lobeline hydrochloride sale extension formulae given in [2]. In particular, while is the standard complex hyperbolic space and the CR-boundary is the Heisenberg group, Frank–González–Monticelli–Tan gave an answer to the extension problem in [9]. In the general case, the energy extension formulae is as follows:
    This paper is organised as follows. In Section 2, we introduce the geometric setting and previous results of the spectral and scattering theory in this background. In section 3, we talk about the case that the boundary CR-structure has positive (nonnegative) Webster scalar curvature and prove Part (a) and (b) of Theorem 1.2. In Section 4, we prove the two energy extension formulae given in Theorem 1.3 and Part (c) of Theorem 1.2.
    Geometric setting Suppose X is a compact manifold of complex dimension , with boundary . Denote by the interior of X. Let be a boundary defining function, i.e. in , on M and for all . We make two assumptions on X and ρ throughout the following paper.
    Nonegative CR-boundary Throughout this section, we assume (A1)–(A2) and Then is (approximate) ACHE and the one-form induces a pseudo-Hermitian structure which is pseudo-Einstein on the boundary M.
    Energy identity Throughout of this section, we fix with . Assume (A1)–(A4) and Hence by Lemma 3.3, ρ is strictly plurisubharmonic all over X. Now we have two Kähler metrics and , associated with Kähler forms and respectively. In local holomorphic coordinates, In our convention, we take More over, near the boundary we have a second set of frame with coframe , which are compatible with the boundary CR-structure on each level set , such that We defined three functions:
    Similar as the energy identity for adapted metric measure space in [2], we have a second energy identity here such that the interior energy has no zero order term.
    Introduction The molecular mechanisms underlying pathogenesis of Alzheimer's disease (AD) involve sequential proteolysis of the large, transmembrane, amyloid precursor protein (APP) by the two aspartic proteinases, β- and γ-secretase. As a result, APP cleavage produces the 39–42 amino acid amyloid-β peptide (Aβ) and releases a soluble ectodomain APPβ (sAPPβ) and a C-terminal fragment, known as AICD (APP intracellular domain) (for review see Refs. [1], [2]). The Aβ peptide, which was originally identified as the main constituent of the inter-neuronal and cerebrovascular plaques characteristic of AD [3], [4], binds to AChE and BChE forming fibrils which are more toxic than aggregates of Aβ alone [5], [6], [7]. The sAPPβ ectodomain was shown to possess some biological activity, including neuroprotection, although 50–100 times less potent than those of the longer sAPPα ectodomain released by the predominant non-amyloidogenic cleavage of APP by α-secretase [8], [9]. Recently, the APP-intracellular domain AICD was shown to play an important role in gene expression and the list of important target proteins which are up- or down-regulated by AICD is rapidly increasing [10], [11]. Both AICD and Aβ have an extensive protein interactome making them important targets for regulation of various cellular functions and of potential therapeutic importance. Similarly, peptides cleaved from AChE also show some regulatory functions, for example in apoptosis, stress-response and neuritogenesis [12]. Although the primary functions of APP and AChE in the organism are distinctly different, there are certain similarities and relationships in their metabolism. Both APP and AChE are widespread membrane-bound proteins which possess important roles in brain development and function. More recent studies suggest that there might yet be some additional roles for them such as cell proliferation and apoptosis, or cell-cell interactions [13], [14]. AChE and APP have several splice variants which differ in length and cell location and possess different cellular functions. Moreover, cells produce soluble forms of these proteins and their secretion can be activated by various stimuli including cholinergic agonists [13], [14], [15]. Like sAPP, soluble forms of AChE are believed to play special roles in neurogenesis, synaptogenesis and neuroprotection [16], [17], [18]. Both APP and cholinesterases (AChE and BChE) were also recently shown to be important for brain development during embryogenesis [19], [20].