一个含临界指数的拟线性椭圆型方程的注记

研究了如下的拟线性椭圆型方程:△pu+uq+λup*-1=0,u∈W1o,p(Ω), (1λ)其中,Ω2是RN中具有光滑边界的有界区域,△pu=div( |▽u|p-2▽u),N≥3,2≤p＜N,0＜q＜1,p*=NP/N-P.设λ*(Ω,p,q)是拟线性椭圆型方程(1λ)可解的参数集的上确界.运用变分方法,在不要求具有对称性质的一般区域Ω上得到了λ*(Ω,p,q)的一个可以精确计算的下界.     

Residue matrices and related properties in two-variable reactance functions

In this paper, the characterization of a two-variable reactance polynomial φ(λ,μ) is given in terms of the residue matrices of a single variable reactance matrix, Y(λ). Specificially, if Y(λ) is expressed in terms of its partial fraction as Y(λ)=λH_∞+Σ$\text{H}{}_{\mathrm{i}}\text{+jβ}{}_{\mathrm{i}}\text{λ+jω}{}_{\mathrm{i}}$+G where the residue matrices in general are p.s.d. Hermitian, then the ranks of these residue matrices are fixed in relation to the construction of φ(λ,μ) as the determinant of the two-variable reactance matrix μ1+Y(λ). Three theorems concerning these ranks—one each corresponding to the finite poles, poles at ∞ and the behaviour at λ=0 of Y(λ) are stated and proved. Several properties following from these theorems are studied. Also, implications of these theorems from a network theoretic point of view, like the minimum number of gyrators required to synthesize Y(λ) to yield the specific type of φ(λ,μ) etc., are studied. In the sequel, the concept of “generalized compact pole conditions” is introduced. Finally, these results are applied for the generation of two-variable reactance functions and matrices.     

On convergence criteria for jarratt's method

Let f(χ) together with its first two derivatives be continuous in the domain D and additionally let χ_M?D be an extremum (or turning point) of this function. Also, let χ_n+1 = T (χ_n,χ_n-1,χ_n-2) be Jarratt's Method for computing the extremum (or turning point) of a function. Criteria are demonstrated which insure that, for any triple of initial assumptions (χ₁,χ₀,χ_-1)?D, Jarratt's Method, converges to the extremum of f(χ), and that from and after some n = N₀, the rate of convergence of this method increases steadily, finally becoming unbounded when the solution χ_M is attained.     

On recursive averaging processes and Hilbert space extensions of the contraction mapping principle

If T maps a convex domain D_T into itself, and if {ω_n} is a real sequence with range in (0, 1] then the recursive averaging process,

$\text{X}{}_{\mathrm{n+1}}\text{=(1?omega;}{}_{\mathrm{n}}\text{)}\text{X}{}_{\mathrm{n}}\text{+ω}{}_{\mathrm{n}}\text{+ω}{}_{\mathrm{n}}\text{T}\text{x}{}_{\mathrm{n}}\text{,}\text{x}{}_0\text{=ξ?D}{}_{\mathrm{T}}$

generates a sequence {x?_n}; with range in D_T. Under suitable conditions on D_T, T and {ω_n} the sequence {x?_n} will converge in some sense to a fixed point of T. We prove that if D_T is a closed convex subset of a complex Hilbert space H, if Tω = (1 ? ω) I + ωT is a strict contraction for some ω ? (0, 1], and if {ω_n} satisfies the conditions,

$\text{ω}{}_{\mathrm{n}}\text{→ 0}$

and

$\text{∑}\text{n=0}\text{∞}\text{ω}{}_{\mathrm{n}}\text{=∞}$

then, for arbitrary ξ ? D_T, {x?_n} converges strongly to (the unique) fixed point of T. We also prove that if D_T and {ω_n} satisfy the foregoing conditions, if T has at least one fixed point, and if Tω is non-expansive for some ω ? (0, 1], then for all ξ ? D_T, {x?_n} converges at least weakly to some fixed point of T. Finally, we apply these results to linear equations involving bounded normal operators and obtain an extension of the classical Neumann operator series.     

Alternative convergence criteria for iterative methods of solving nonlinear equations

Let χ_m+1=T(χ_m) or even χ_m+1=T(χ_m,χ_m?1, …, χ_m?q), m=1,2,3 … be an iteration method for solving the nonlinear problem F(χ)=0, where F(χ) and its derivatives possess all of the properties required by T(χ_m). Then if it can be established that for the problem at hand ∥F(χ_m+1)∥?β_m∥F(χ_m)∥, ? m > M₀ (M₀<∞) and 0?β_m<1 , definitions are established and theorems proven concerning convergence, uniqueness and bounds on the error after ‘m’ successive iterations of a new approach to convergence properties T(χ_m). These charateristics are referred to as “alternate” (local, global) convergence properties and none of the proofs given are restricted to any specific type of method such as, e.g. contraction mapping types. Application of results obtained are illustrated using Newton's method as well as the general concept of Newton-like methods.     

Gossips and telegraphs

Suppose we have a group of n people, each possessing an item of information not known to any of the others and that during each unit of time each person can send all of the information he knows to at most other people. Further suppose that each of at most k other people can send all of the information they know to him. Determine the length of time required, f(n, k), so that all n people know all of the information. We show f(n, k) = ?log_k+1n?.We define g(n, k) analogously except that no person may both send and receive information during a unit time period. We show ?log_k+1n?≤g(n, k)≤2?log_k+1n? in general and further show that the upper bound can be significantly improvea in the cases k = 1 or 2. We conjecture g(n, k) = b_k log_k+1n+0(1) for a function b_k we determine.     

Periodic solutions to systems of reaction-diffusion equations

In this paper, necessary and sufficient conditions are derived for the existence of temporally periodic “dissipative structure” solutions in cases of weak diffusion with the reaction rate terms dominant in a generic system of reaction-diffusion equations ?c_i/?t = D_i?²c_i+Q_i(c), where the enumerator index i runs 1 to n, c_i = c_i(x, t) denotes the concentration or density of the ith participating molecular or biological species, D_i is the diffusivity constant for the ith species and Q_i(c), an algebraic function of the n-tuple c = (c₁,\3., c_n), expresses the local rate of production of the ith species due to chemical reactions or biological interactions.     

Mathematical models of information system use

This report presents the results from a study of mathematical models relating to the usage of information systems. For each of four models, the papers developed during the study provide three types of analyses: reviews of the literature relevant to the model, analytical studies, and tests of the models with data drawn from specific operational situations. (1) The Cobb-Douglas model: x₀ = ax₁^bx₂^(1?b).This classic production model, normally interpreted as applying to the relationship between production, labor, and capital, is applied to a number of information related contexts. These include specifically the performance of libraries, both public and academic, and the use of information resources by the nation's industry. The results confirm not only the utility of the Cobb-Douglas model in evaluation of the use of information resources, but demonstrate the extent to which those resources currently are being used at significantly less than optimum levels. (2) Mixture of Poissons:

$\text{χ}{}_0\text{=}\text{∑}\text{i=0}\text{n}\text{i}\text{∑}\text{j=0}\text{p}\text{n}{}_{\mathrm{j}}\text{e}{}^{\mathrm{mj}}\text{(m}{}_{\mathrm{j}}\text{)′/i!}$

where x₀ is the usage and (n_j,m_j),j = 0 to p, are the p + 1 components of the distribution. This model of heterogeneity is applied to the usage of library materials and of thesaurus terms. In each case, both the applicability and the analytical value of the model are demonstrated. (3) Inverse effects of distance: x = a e^?md if c(d) = rdx = ad^?m if c(d) = r log(d).These two models reflect different inverse effects of distance, the choice depending upon the cost of transportation. If the cost,c(d), is linear, the usage is inverse exponential; if logarithmic, the usage is inverse power. The literature that discusses the relationship between usage of facilities and the distance from them is reviewed. The models are tested with data from the usage of the Los Angeles Public Library, both Central Library and branches, based on a survey of 3662 users. (4) Weighted entropy:

$\text{S(x}{}_1\text{,x}{}_2\text{,...,x}{}_{\mathrm{n}}\text{)= -}\text{∑}\text{i=1}\text{n}\text{r(x}{}_{\mathrm{i}}\text{P(x}{}_{\mathrm{i}}\text{)}\text{log}\text{(p(x}{}_{\mathrm{i}}\text{)).}$

This generalization of the “entropy measure of information” is designed to accommodate the effects of “relevancy”, as measured by r(x), upon the performance of information retrieval systems. The relevant literature is reviewed and the application to retrieval systems is considered.     

A test for robust Hurwitz stability of convex combinations of complex polynomials

In this paper we present a method for testing the Hurwitz property of a segment of polynomials (1−λ)p₀(s)+λp₁(s), where λ∈[0,1] and p₀(s) and p₁(s) are nth-degree polynomials with complex coefficients. The method consists in constructing a parametric Routh-like array with polynomial entries and generating Sturm sequences for checking the absence of zeros of two real λ-polynomials of degrees 2 and 2n in the interval (0,1). The presented method is easy to implement. Moreover, it accomplishes the test in a finite number of arithmetic operations because it does not invoke any numerical root-finding procedure.     

一类含Hardy项的三维Kirchhoff型问题的两个正解

研究如下的三维Kirchhoff型问题{-(a+b∫Ω| ▽u | 2dx)△u=| u|q-1u+λ |u|p-2u/|x|s, x∈Ω,u=0, x∈(a)Ω,其中,Ω是R3中具有光滑边界的有界区域,0∈Ω,0＜q＜1,0≤s＜1,4＜p＜2*(s)=2(3-s),a,b,λ＞0.运用变分方法,证明当λ＞0足够小时,这一方程至少有2个正解.     

Partial derivative properties of multivariable cascaded networks1

Some partial derivative properties of the impedance function of a resistively-terminated network which is a cascade of m lossless two-ports of variables p₁ to p_m dre established. Those cases where some of the lossless two-ports are lowpass or highpass ladder networks with all of their transmission zeros at the origin or at infinity are also examined. Necessary and sufficient conditions involving partial derivatives and under which an m-variable reactance or positive real function can be realized as the impedance function of a p_i-variable lowpass or highpass ladder network, with all of its transmission zeros at p_i = 0 or p_i = ∞, and terminated in a reactance or positive real impedance function in the other variables are derived.     

Stability criteria for certain classes of nonlinear difference equations

The general mth order difference equation X(n+m)+a₁X(n+m?1)+…+a_mX(n) = F[n,X(n),…,X(n+m?1)] is considered. The stability properties of its solutions are studied using the discrete form of Liapunov's direct method. A quadratic form is selected as a possible Liapunov function V(n,X) and a scheme is developed for determining appropriate conditions on this function to insure that its total difference ΔV(n,X) is negative semi-definite or negative definite with respect to the difference equation. The approach is applied to the fourth-order difference equation in full detail to illustrate the method for determining the conditions which imply either uniform stability or uniform asymptotic stability and specific results are obtained. Several comments on, and extensions of, the work done by Puri and Drake for the cases m = 2 and m = 3 are presented.The results of the present approach in the homogeneous case where F[n,X(n),…,X(n+m?1)] = 0 are compared with the usual Schur-Cohn criteria and are shown to be at least as good.     

Mineralogy of the deep lower mantle in the presence of H2O

Understanding the mineralogy of the Earth''s interior is a prerequisite for unravelling the evolution and dynamics of our planet. Here, we conducted high pressure-temperature experiments mimicking the conditions of the deep lower mantle (DLM, 1800–2890 km in depth) and observed surprising mineralogical transformations in the presence of water. Ferropericlase, (Mg, Fe)O, which is the most abundant oxide mineral in Earth, reacts with H₂O to form a previously unknown (Mg, Fe)O₂H_x (x ≤ 1) phase. The (Mg, Fe)O₂H_x has a pyrite structure and it coexists with the dominant silicate phases, bridgmanite and post-perovskite. Depending on Mg content and geotherm temperatures, the transformation may occur at 1800 km for (Mg_0.6Fe_0.4)O or beyond 2300 km for (Mg_0.7Fe_0.3)O. The (Mg, Fe)O₂H_x is an oxygen excess phase that stores an excessive amount of oxygen beyond the charge balance of maximum cation valences (Mg²⁺, Fe³⁺ and H⁺). This important phase has a number of far-reaching implications including extreme redox inhomogeneity, deep-oxygen reservoirs in the DLM and an internal source for modulating oxygen in the atmosphere.     

(1+λu)-Constacyclic codes over Fp[u]/〈u〉

Motivated by the work in [1] of Abualrub and Siap (2009), we investigate (1+λu)-constacyclic codes over F_p[u]/〈u^m〉 of an arbitrary length, where λ is a nonzero element of F_p. We find the generator polynomials of (1+λu)-constacyclic codes over F_p[u]/〈u^m〉, and determine the number of (1+λu)-constacyclic codes over F_p[u]/〈u^m〉 for a given length, as well as the number of codewords in each such code. Some optimal linear codes over F₃ and F₅ are constructed from (1+λu)-constacyclic codes over F_p+uF_p under a Gray map.     

Some New Convergence Conditions for Jacobi's Method

Let Aχ = b be a linear system or equations in which A is an n × n matrix and both χ and b are n dimensional column vectors. If in addition A is row diagonally predominant (i.e. for any i, ¦:a_ii¦>¦:a_ij¦: for all 1?j?n,i≠j), then criteria are developed to show how to ascertain when such matrix problems may be successfully solved using Jacobi's iterative method.     

On the uniqueness of the minimal solution to the matrix polynomial equation A(λ)X(λ)+Y(λ)B(λ)=C(λ)

The uniqueness of the minimal solution of the matrix equation A(λ)X(λ)+Y(λ)B(λ)=C(λ) is discussed in this paper. It is proved that Barnett's condition for uniqueness can be obtained when only A(λ) or B(λ) (not necessarily both) is regular. Furthermore, it is shown that if such a minimal solution exists, it is not unique if and only if both matrices are nonregular.     

Perturbation solutions of the Carson–Cambi equation

The periodic differential equation (1+ε cos t)y&#x030B; + py = 0, hereby termed the Carson–Cambi equation, is the simplest second-order differential equation having a periodic coefficient associated with the second derivative. Provided |ε|<1, which is the case we examine, then the differential equation is a Hill's equation and thus possesses regions of stability and instability in the p–ε plane. Ordinary perturbation theory is employed to obtain the stable (periodic) solutions to ε³. Two-timing theory is employed to obtain solutions for values of k near the critical points k = ±$\text{1}\text{2}$, ±$\text{3}\text{2}$, ±$\text{5}\text{2}$. Three-timing is employed to extend the solution near k = ±$\text{1}\text{2}$. The solutions of the Carson–Cambi equation are compared with the solutions of the corresponding Mathieu equation.     

An equation of state for a system with a single type of transformation

Based on theory of a previous paper, the writer has developed an equation of state for a system with a single type of transformation. This equation is of the form

$\text{h=A+Bv+Cp+Dpv?T(E+Fv+Gp+Hpv)}$

where h = ε + pv is the total heat, p the pressure, v the specific volume, T the temperature, and p, v, T are considered independent variables. A, B, C, etc., are constants for the system. The latent eat at constant (p, T) is given by

$\text{λp,T=(v}{}_2\text{?v}{}_1\text{)(}\text{?h}\text{?v}\text{)P,T= (v}{}_2\text{?v}{}_1\text{)[(B?TF)+p(D?TH)]}$

. These equations are checked with data on saturated and superheated ammonia, and the agreement is good to within a few tenths of a per cent. Also, checks with data on saturated and superheated steam show agreement within several per cent.     

Protective effect of CD73 inhibitor α, β-methylene ADP against amyloid-β-induced cognitive impairment by inhibiting adenosine production in hippocampus

BackgroundAlzheimer's disease (AD) is a chronic, progressive neurodegenerative disease. Recent studies have reported the close association between cognitive function in AD and purinergic receptors in the central nervous system. In the current study, we investigated the effect of CD73 inhibitor α, β-methylene ADP (APCP) on cognitive impairment of AD in mice, and to explore the potential underlying mechanisms.ResultsWe found that acute administration of Aβ_1–42 (i.c.v.) resulted in a significant increase in adenosine release by using microdialysis study. Chronic administration of APCP (10, 30 mg/kg) for 20 d obviously mitigated the spatial working memory impairment of Aβ_1–42-treated mice in both Morris water maze (MWM) test and Y-maze test. In addition, the extracellular adenosine production in the hippocampus was inhibited by APCP in Aβ-treated mice. Further analyses indicated expression of acetyltransferase (ChAT) in hippocampus of mice of was significantly reduced, while acetylcholinesterase (AChE) expression increased, which compared to model group. We observed that APCP did not significantly alter the NLRP3 inflammasome activity in hippocampus, indicating that anti-central inflammation seems not to be involved in APCP effect.ConclusionsIn conclusion, we report for the first time that inhibition of CD73 by APCP was able to protect against memory loss induced by Aβ_1–42 in mice, which may be due to the decrease of CD73-driven adenosine production in hippocampus. Enhancement of central cholinergic function of the central nervous system may also be involved in the effects of APCP.How to cite: Song W, Tang Y, Wei L, et al. Protective effect of CD73 inhibitor α, β-methylene ADP against amyloid-β-Induced cognitive impairment by inhibiting adenosine production in hippocampus. Electron J Biotechnol 2020;48. https://doi.org/10.1016/j.ejbt.2020.09.002     

Fixed point properties of the binomial function

Fixed point properties of the binomial function

are developed. It is shown that for any 1 < L < N, T^L_Nhas a unique fixed point p? in (0, 1), and that for large N, the fixed point is L/N. This has application to signal detection schemes commonly used in communication systems. When detecting the presence or absence of a signal with an initial false alarm probability p_FAand an initial detection probability p_D, then T^L_N(p_FA) < p_FAand T^L_N(p_D) > p_Dif, and only if, p_FA < p? < p_D. When this condition is satisfied, as N → ∞, T^L_N(p_FA) → 0 and T^L_N(p_D → 1.