A note on continuous fractional wavelet transform in ℝn

In this paper, we study the continuous fractional wavelet transform (CFrWT) in [Formula: see text]-dimensional Euclidean space [Formula: see text] with scaling parameter [Formula: see text] such that [Formula: see text]. We obtain inner product relation and reconstruction formula for the CFrWT depending on two wavelets along with the reproducing kernel function, involving two wavelets, for the image space of CFrWT. We obtain Heisenberg’s uncertainty inequality and Local uncertainty inequality for the CFrWT. Finally, we prove the boundedness of CFrWT on the Morrey space [Formula: see text] and estimate [Formula: see text]-distance of the CFrWT of two argument functions with respect to different wavelets.

Generalized second law and universal relations of cosmological black hole

The objective of this paper is to investigate the validity conditions for the generalized second law of thermodynamics, and the universal relations for multi-horizon dynamical spacetime. It is found that there are three horizons of McVittie universe termed as event horizon, cosmological apparent horizon, and virtual horizon. The mass-dependent and mass-independent area product relations are formulated in terms of areas of the dynamical event horizon, cosmological horizon and virtual horizon. It is noted that whereas the area sum relation is mass independent, the area product relation is explicitly mass dependent. Moreover, we have also analyzed and listed explicit mass-independent and mass-dependent relations.

Geometric Programming with Discrete Variables Subject to Max-Product Fuzzy Relation Constraints

The problem of geometric programming subject to max-product fuzzy relation constraints with discrete variables is studied. The major difficulty in solving this problem comes from nonconvexity caused by these product terms in the general geometric function and the max-product relation constraints. We proposed a 0-1 mixed integer linear programming model and adopted the branch-and-bound scheme to solve the problem. Numerical experiments confirm that the proposed solution method is effective.

Logarithmic uncertainty principle, convolution theorem related to continuous fractional wavelet transform and its properties on a generalized Sobolev space

The continuous fractional wavelet transform (CFrWT) is a nontrivial generalization of the classical wavelet transform (WT) in the fractional Fourier transform (FrFT) domain. Firstly, the Riemann–Lebesgue lemma for the FrFT is derived, and secondly, the CFrWT in terms of the FrFT is introduced. Based on the CFrWT, a different proof of the inner product relation and the inversion formula of the CFrWT are provided. Thereafter, a logarithmic uncertainty relation for the CFrWT is investigated and the convolution theorem related to the CFrWT is established using the convolution of the FrFT. The CFrWT on a generalized Sobolev space is introduced and its important properties are presented.

Fin-set: A syntactical definition of finite sets

We state Fin-set, by which one founds the notion of finite sets in a syntactical way. Any finite set {a1, a2,..., an} is defined as a well formed term of the form S(a1 + (a2 + (??? + (an?1 + an)???))), where + is a binary and S a unary operational symbol. Related to the operational symbol the term-substitutions (1) are introduced. Definition of finite sets is called syntactical because by two algorithms Set-alg and Calc one can effectively establish whether any given set-terms are equal or not equal. All other notions related to finite sets, like ?, ordered pair, Cartesian product, relation, function, cardinal number are defined as terms as well. Each of these definitions is recursive. For instance, ? is defined by x ? S(a1) iff x = a1 x ? S(a1 + ???+ an) iff x = a1 or x ? S(a2 + ???+ an) x/? ? (? denotes the empty set).

Serine metabolism in human pregnancy

Serine plays an important role in intermediary metabolism as a source of one carbon pool for nucleotide biosynthesis, as a precursor for glycine and glucose, and as a contributor to cysteine biosynthesis. A unique serine-glycine cycling between the liver and the placenta has been demonstrated in the sheep fetus. We hypothesized that, because of serine's role in growth and development, significant changes in serine metabolism will occur in pregnancy with advancing gestation. The rate of appearance (Ra) of serine and its metabolism were quantified in healthy women longitudinally through pregnancy with a [2-15N13C]serine tracer. The contribution of serine N to urea and the rate of oxidation of serine were measured using the precursor-product relation. Plasma serine concentrations and serine Ra were lower in pregnant (P) women, in both early and late gestation, compared with nonpregnant (NP) women [plasma serine: NP, 113 ± 24.5; P early, 71.9 ± 6.2; P late, 68.5 ± 9.6 μmol/l; serine Ra: NP ( n = 7), 152.9 ± 42.8; P early ( n = 12), 123.7 ± 21.5; P late ( n = 8), 102.8 ± 18.2 μmol · kg−1 · h−1]. Serine contributed ∼6% to urea N and 15–20% to the plasma glycine pool, and oxidation of serine represented ∼8% of Ra. There was no significant difference between P and NP subjects. Glucose infusion, at 3 mg · kg−1 · min−1in P subjects, resulted in a decrease in serine Ra and an increase in oxidation. The decrease in serine turnover in pregnancy may represent a decrease in α-amino nitrogen turnover related to a decreased rate of branched-chain amino acid transamination and caused by pregnancy-related hormones aimed at nitrogen conservation and accretion.