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Let M n = diag [ 3 p n , 3 q n ] with p n , q n ⩾ 1 for all n ⩾ 1 and let D = { (0 , 0) t , (1 , 0) t , (0 , 1) t } . One can generate a Borel probability measure μ { M n } , D = δ M 1 − 1 D ∗ δ (M 2 M 1) − 1 D ∗ δ (M 3 M 2 M 1) − 1 D ∗ ⋯. Such measure μ { M n } , D is called a Moran Sierpinski-type measure. It is known Deng et al (Acta Math. Sin. submitted) that the associated Hilbert space L 2 (μ { M n } , D) has an exponential orthonormal basis. In this paper, we first characterize all the maximal exponential orthogonal sets for L 2 (μ { M n } , D) . For such a maximal orthogonal set, we then give some sufficient conditions to determine whether it is an orthonormal basis of L 2 (μ { M n } , D) or not. [ABSTRACT FROM AUTHOR]

G (1-5)-N[H.sub.2], G (1-7)-N[H.sub.2], and G (1-9) are the active fragments of ghrelin. The aim of this study was to investigate the antinociceptive effects, their ability to cross the blood-brain barrier, and the receptor mechanism(s) of these fragments using the tail withdrawal test in male Kunming mice. The antinociceptive effects of these fragments (2, 6, 20, and 60 nmol/mouse) were tested at 5,10, 20, 30, 40, 50, and 60 min after intravenous (i.v.) injection. These fragments induced dose- and time-related antinociceptive effects relative to saline. Using the near infrared fluorescence imaging experiments, our results showed that these fragments could cross the brain-blood barrier and enter the brain. The antinociceptive effects of these fragments were completely antagonized by naloxone (intracerebroventricular, i.c.v.); however, naloxone methiodide (intraperitoneal, i.p.), which is the peripheral restricted opioid receptor antagonist, did not antagonize these antinociceptive effects. Furthermore, the GHS-R1[alpha] antagonist [[D-Lys.sup.3]]-GHRP-6 (i.c.v.) completely antagonized these antinociceptive effects, too. These results suggested that these fragments induced antinociceptive effects through central opioid receptors and GHS-R1[alpha]. In conclusion, our studies indicated that these active fragments of ghrelin could cross the brain-blood barrier and enter the brain and induce antinociceptive effects through central opioid receptors and GHS-R1[alpha] after intravenous injection. Key words: G, GHS-R1[alpha], opioid receptors, antinociception, blood-brain barrier. G (1-5)-N[H.sub.2], G (1-7)-N[H.sub.2] et G (1-9) representent les fragments actifs de la ghreline. Cette etude portait sur les effets antinociceptifs, la capacite de franchir la barriere hemo- encephalique et le(s) mode(s) d'action des recepteurs de ces fragments a l'aide du test de retrait de la queue chez des souris Kunming males. Nous avons etudie les effets antinociceptifs de ces fragments (2, 6, 20, 60 nmol/souris) 5,10, 20, 30, 40, 50 et 60 min apres leur injection intraveineuse (i.v.). Ces fragments entrainaient des effets antinociceptifs proportionnels a la dose et a la duree par rapport a la solution saline. Les resultats de nos experiences d'imagerie par la fluorescence proche infrarouge (NIR) ont montre que ces fragments pouvaient franchir la barriere hemo-encephalique et penetrer dans le cerveau. La naloxone par voie intracerebrovasculaire (i.c.v.) entrainait une inhibition complete des effets antinociceptifs de ces fragments (i.v.). Cependant, l'administration intraperitoneale (i.p.) de methiodide de naloxone, l'antagoniste des recepteurs opioides restreint a la peripherie, n'entrainait pas d'inhibition de ces effets antinociceptifs. Pourtant, le [[D-Lys.sup.3]]-GHRP-6, antagoniste du GHS- R1[alpha], entrainait aussi une inhibition complete de ces effets antinociceptifs par voie i.c.v. Ces resultats laissent entendre que ces fragments (i.v.) auraient entraine des effets antinociceptifs par l'intermediaire des recepteurs opioides et des recepteurs GHS- R1[alpha] centraux. En conclusion, nos etudes montraient que ces fragments actifs de la ghreline pouvaient franchir la barriere hemo-encephalique et penetrer dans le cerveau pour engendrer des effets antinociceptifs par l'intermediaire des recepteurs opioides et GHS-R1[alpha] centraux apres leur injection par voie intraveineuse. [Traduit par la Redaction] Mots-cles: G, GHS-R1[alpha], recepteurs opioides, antinociception, barriere hemo-encephalique., 1. Introduction Pain is a major public health issue that affects many people and imposes high costs (Military Medicine 2016). The clinical use of many analgesics is hampered by its [...]

Let p, q, N ≥ 2 {N\geq 2} be three positive integers and let D = { 0 , 1 , … , N - 1 } ⊕ N p ⁢ { 0 , 1 , … , N - 1 } {D=\{0,1,\ldots,N-1\}\oplus N^{p}\{0,1,\ldots,N-1\}} be a product form digit set. It is well known that if q ∤ p {q\nmid p} , then the self-similar measure μ N q , D {\mu_{N^{q},D}} generated by the iterated function system { ( N q ) - 1 ⁢ ( x + d ) } d ∈ D , x ∈ ℝ {\{(N^{q})^{-1}(x+d)\}_{d\in D,x\in\mathbb{R}}} is a spectral measure with a spectrum Λ ⁢ ( N q , C ) = { ∑ i = 0 finite c i ⁢ N q ⁢ i : c i ∈ C } , \Lambda(N^{q},C)=\Bigg{\{}\sum_{i=0}^{\text{finite}}c_{i}N^{qi}:c_{i}\in C% \Bigg{\}}, where C = N q - p - 1 ⁢ D {C=N^{q-p-1}D} . In this paper, based on the properties of cyclic groups in number theory, we give some conditions on real number t under which the scaling set t ⁢ Λ ⁢ ( N q , C ) {t\Lambda(N^{q},C)} is also a spectrum of μ N q , D {\mu_{N^{q},D}} . [ABSTRACT FROM AUTHOR]

Let 0 < ρ < 1 and let { a j , b j , n j } j = 1 ∞ be a sequence of positive integers with an upper bound. Associated with them, there exists a unique Borel probability measure μ ρ , { 0 , a j , b j } , { n j } generated by the following infinite convolution of discrete measures: μ ρ , { 0 , a j , b j } , { n j } = δ ρ n 1 ⁢ { 0 , a 1 , b 1 } ∗ δ ρ n 1 + n 2 ⁢ { 0 , a 2 , b 2 } ∗ δ ρ n 1 + n 2 + n 3 ⁢ { 0 , a 3 , b 3 } ∗ ⋯ , where gcd ⁡ (a j , b j) = 1 for all j ∈ ℕ . In this paper, we show that L 2 ⁢ (μ ρ , { 0 , a j , b j } , { n j } ) admits an exponential orthonormal basis if and only if the following two conditions are satisfied: (i) { a j , b j } ≡ { ± 1 } ⁢ (mod ⁢ 3) for all j ≥ 1 ; (ii) there exists a natural number r such that ρ - r ∈ 3 ⁢ ℕ and n j ∈ r ⁢ ℕ for all j ≥ 2 . [ABSTRACT FROM AUTHOR]