and Applied Analysis 3 capacity of the prey population x t is b/a, so it is meaningful that the economical threshold h is less than b/a. Thus, throughout this paper, we set up the following two assumptions: A1 D e < a b , A2 h ≤ b a . 2.1 From the biological point of view, it is reasonable that system 1.1 is considered to control the prey population in the biological meaning space { x, y : x ≥ ...

Let f(x) = ax+bx+c ∈ Z[x] be a quadratic polynomial with a ̸≡ 0 mod p. Take z ∈ Fp and let Oz = {fi(z)}i∈Z+ be the orbit of z under f , where fi(z) = f(fi−1(z)) and f0(z) = z. For M < |Oz|, We study the diameter of the partial orbit OM = {z, f(z), f2(z), . . . , fM−1(z)} and prove that there exists c1 > 0 such that diam OM & min { Mp c1 , 1 log p M 4 5p 1 5 ,M 1 13 log logM } . For a complete or...

-’We present a theory of quadratic time-frequency (TF) energy distributions that satisfy a covariance property and generalized marginal properties. The theory coincides with the characteristic function method of Cohen and Earaniuk in the special case of ‘‘conjugate operators.” 1 I N T R O D U C T I O N A N D O U T L I N E Important c1.asses of quadratic time-frequency representations (QTFRs), s...

-’We present a theory of quadratic time-frequency (TF) energy distributions that satisfy a covariance property and generalized marginal properties. The theory coincides with the characteristic function method of Cohen and Earaniuk in the special case of ‘‘conjugate operators.” 1 I N T R O D U C T I O N A N D O U T L I N E Important c1.asses of quadratic time-frequency representations (QTFRs), s...

In [P] R. Pellikaan introduced a two variable zeta-function Z(t, u) for a curve over a finite field Fq which, for u = q, specializes to the usual zeta-function and he proved, among other things, rationality: Z(t, u) = (1 − t)−1(1 − ut)−1P (t, u) with P (t, u) ∈ Z[t, u]. We prove that P (t, u) is absolutely irreducible. This is motivated by a question of J. Lagarias and E. Rains about an analogo...

A basic question in probability, statistics, and machine learning is the following: given a random variable Z with expectation E[Z], how likely is Z to be close to its expectation? And more precisely, how close is it likely to be? With that in mind, these notes give a few tools for computing bounds of the form P(Z ≥ E[Z] + t) and P(Z ≤ E[Z]− t) (1) for t ≥ 0. Our first bound is perhaps the most...

where y = [yl y~ " ' " y~]' has covariance mat r ix r and is uncorre la ted wi th z = [zl z2 . z~]' which has covar iance mat r ix A, so t h a t z¢ is uncorre la ted wi th zi , 1 < i _~ j < p. T h e variable y~ is called the common-fac tor c o m p o n e n t of x~ and var(y~) -1 -~ is the communa l i ty of x¢ while z~ is the specific fac tor of x~ and var(z~) = ~ is the uniqueness of x¢ . Le t p...

In this paper, we mainly discuss the nonuniform average sampling and reconstruction in multiply generated shift-invariant subspaces \[ V_{p,q}(\Phi_r) = \bigg\{ \sum_{k_{1} \in \mathbf{Z}} \sum_{k_{2} \mathbf{Z}^{d}} c^T(k_{1},k_{2}) \Phi_r(\,\cdot-k_{1},\,\cdot-k_{2}): (c(k_{1},k_{2}))_{(k_{1},k_{2}) \mathbf{Z} \times \big( \ell^{p,q}(\mathbf{Z} \mathbf{Z}^d) \big)^r \bigg\} \] of mixed Lebesg...

C o n d e n s a t i o n o f i s a t i n w i t h 4 a m i n o a c e t o p h e n o n e g a v e a q u a n t i t a t i v e y i e l d o f i s a t y l i d e n e j 9 a c e t y l a n i l i n e ( 1 ) . C o n d e n s a t i o n o f 1 w i t h a r o m a t i c a l d e h y d e s y i e l d e d t h e c o r r e s p o n d i n g i s a t y l i d e n e a z o m e t h i n e c h a l c o n e s ( 2 a f ) . I n t e r a c t...

machine of AMINE Here, we review the definition of the abstract machine formally: 42 Definition 3.3.2 (Notations for maps) • We use [ ] as an empty map. • Let ψ be a map. We use the following notation: ψ[x 7→ a](z) def =  a (z is x) ψ(z) (otherwise). When ψ(x) is undefined, we use the following notation: ψ[x 7→ ⊥](z) def =  undefined (z = x) ψ(z) (otherwise). We may also write ψ(x) = ⊥ when...