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A general system **of** q-orthogonal **polynomials** is dened by means **of** its three-term recur- rence relation. This system encompasses many **of** **the** known families **of** q-**polynomials**, among them **the** q-analog **of** **the** **classical** orthogonal **polynomials**. **The** **asymptotic** density **of** **zeros** **of** **the** system is shown to be a simple **and** compact expression **of** **the** parameters which char- acterize **the** **asymptotic** behavior **of** **the** coecients **of** **the** recurrence relation. This result is applied to specic classes **of** **polynomials** known by **the** names q-Hahn, q-Kravchuk, q-Racah, q-Askey & Wilson, Al Salam-Carlitz **and** **the** celebrated q-little **and** q-big Jacobi.

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It can be shown (see e.g., [3,8,25]) that **the** only families satisfying **the** above deﬁnition are **the** Hermite, **Laguerre**, Jacobi, **and** Bessel **polynomials**. Nevertheless there are other **properties** characterizing such families **and** that can be used to deﬁne **the** **classical** OPS. **The** oldest one is **the** so called Hahn characterization—unless this was ﬁrstly observed **and** proved for **the** Jacobi, **Laguerre**, **and** Hermite **polynomials** by Sonin in 1887. In [12], Hahn proved **the** following, Theorem 1.2 (Sonin–Hahn [12,19]). A given sequence **of** orthogonal **polynomials** (P n ) n , is a **classical** sequence if

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L (α) n (x) = 2 − 1 π − 1/2 e x/2 ( − x) − α/2 − 1/4 n α/2 − 1/4 e 2( − nx) 1 / 2 1 + O n − 1/2 . (4) This relation holds for x in **the** complex plane cut along **the** positive real semiaxis; both ( − x) − α/2 − 1/4 **and** ( − x) 1/2 must be taken real **and** positive if x < 0 . **The** bound **of** **the** remainder holds uniformly in every closed domain which does not overlap **the** positive real semiaxis.

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We describe **the** general theory **of** diffusion processes, which contains as a particular case **the** solutions **of** stochastic differential equations. **The** idea **of** **the** theory is to construct explicitly **the** generator **of** **the** Markov process using **the** so-called scale function **and** **the** speed measure. We also explain how **the** theory **of** orthogonal **polynomials** help to study some diffusions. In addition, using **the** theory **of** diffusions, we present **the** Brox model, which is a process in a random environment.

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A matrix versión of this theorem, (see [5,pp. 68]), and Zhedanov has constructed, using the symmetrized Al-Salam-Carlitz polynomials, examples of orthogonal polynomials for a discrete m[r]

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This operation guarantees that **the** substituted **zeros** are inside **the** unit circle, not disturbing **the** minimum-phase property **of** **the** HRTF filter. **The** maximum error in **the** magnitude spectrum is less than 0.3dB when ε is selected as 0.001. **The** same operation can also be applied to a couple **of** **zeros** that are both negative, which will produce new negative quasi-complex **zeros**. **The** number **of** real **zeros** was decreased to 2 for all **of** **the** FIR filters. After this reduction, it is possible to represent **the** FIR filters as a combination **of** 62 complex **zeros** **and** 2 real **zeros**:

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We also test intermediate solutions, where left-shifting is performed but only for a few bits. Fig. 2 shows this new adder that we denote as A2 (A2H for HUB version). Comparing it with A1, **the** A2 design has a special leading zero detector, which detects up to two leading **zeros** at **the** output **of** **the** absolute value circuit. Furthermore, it has a barrel shifter that can perform a one-position right-shifting (in case **of** detecting overflow) **and** left-shifting up to 2-bit positions. This will increase **the** area **and** **the** delay **of** **the** critical path, but it will improve **the** error figures as we will see in section V. In this architecture **the** exponent has to be decremented when left- shifting is performed, **and** therefore underflow could happen. Although it is not depicted in Fig. 2, this situation is detected in **the** design **and** **the** result flushes to zero.

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amount **of** this herbicide molecule being present in cationic form in these soils (Table 4). In **the** case **of** **the** Ultisol soil, **the** higher CEC is de- rived from **the** clay content, **and** in **the** Andisol soil, it is associated with **the** FAH content, which is highly related to **the** adsorption **of** herbicides like MCPA (Iglesias et al., 2009) (Table 5). In general, **the** desorption process did not have a highly significant relationship with any pa- rameter, except for **the** clay content (Table 5). **The** contrast between **the** high correlation **of** **the** desorption percentages **of** herbicides with **the** soil clay content, **and** not with **the** OC content, **and** **the** high correlation between **the** adsorption **of** these herbicides **and** **the** soil OC content (es- pecially **the** HS content), but not **the** soil clay content, indicates **the** presence **of** a certain de- gree **of** hysteresis in **the** adsorption-desorption process. Therefore, **the** adsorption mechanism **of** **the** herbicides in these soils might differ from **the** desorption mechanism (Weber **and** Weil- ing, 1998). However, it is important to consider that **the** estimation **of** **the** desorption process is associated with a high degree **of** uncertainty because **the** adsorption-desorption process re- quires long periods **of** time to equilibrate **and**, in some cases, may take weeks or months to ap- proach a real balance (Ball **and** Roberts, 1991; Weiling **and** Weber, 1998). Subsequently, **the** times for which desorption measurements were taken in this study might have been too short to achieve a realistic balance.

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Our aim here is to express in terms of higher order hypergeometric Lauricella functions the corresponding asymptotic contracted measure of zeros for the sequence {P n (x)}™ =1 to be de[r]

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As previously described, **the** objectification **of** knowledge is a theoretical construct to account for **the** way in which **the** students engage with something in order to notice **and** make sense **of** it. By focusing on **the** students’ phenomenol- ogical mathematical experience, it emphasizes **the** subjective dimension **of** know- ing. But this is only half **of** **the** story. Since we are sociocultural knowers, objec- tification takes also account **of** **the** social **and** cultural dimensions **of** knowing. **The** concept **of** knowledge objectification rests indeed on **the** idea that class- rooms are not merely a bunch **of** external conditions to which **the** students must adapt. Classrooms are rather seen as interactive zones **of** mediated activities con- veying scientific, ethical, aesthetical **and** other culturally **and** historically formed values that **the** students objectify through reflective **and** active participation (Rad- ford, 2008). In these activities, embedded in cultural, historical traditions, **the** students relate not only to **the** objects **of** knowledge (**the** subject-object plane), but also to other students through face-to-face, virtual or potential communica- tive actions (**the** subject-subject plane or plane **of** social interaction).

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for problems with more than a few degrees **of** freedom. Another great advantage **of** **the** semiclassical approximation lies in that it facilitates an intuitive understanding **of** **the** underlying physics, which is usu- ally hidden in blind numerical solutions **of** **the** Schr¨odinger equation. Although semiclassical mechanics is as old as **the** quantum theory itself, **the** field is continuously evolving. There still exist many open problems in **the** mathematical aspects **of** **the** approximation as well as in **the** quest for new effective ways to apply **the** approximation to various physical systems (see, for instance, [1, 2] **and** references therein). In a different vein, applications **of** **the** so-called q-calculus to statistical mechanics have accrued in- creasing interest lately [3]. This q-calculus [4] has its origin in **the** q-deformed harmonic oscillator theory, which, in turn, is based on **the** construction **of** a SU q(2) algebra **of** q-deformed commutation or anti-commutation relations between creation **and** annihilation operators [5–7]. **The** above mentioned applications also employ “deformed information measures” (DIM) that have been applied to different scientific disciplines (see, for example, [3, 8, 9] **and** references therein). DIMs were introduced long ago in **the** cybernetic-information communities by Harvda-Charvat [10] **and** Vadja [11] in 1967-68, being rediscovered by Daroczy in 1970 [12] with several echoes mostly in **the** field **of** image processing. For a historic summary **and** **the** pertinent references see Ref. [13]. In astronomy, physics, economics, biology, etc., these deformed information measures are often called q-entropies since 1988 [9].

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For an historical view **of** this question we refer to a paper by Derbes [ 2 ] which also gives a very complete discussion **of** **the** problem in **the** language **of** **classical** Euclidean geometry, including **the** contribution to this very problem **of** outstanding ﬁ gures such as Maxwell [ 4 ] . **The** historical constructions are extended in this paper even to parabolic orbits ( see also **the** paper [ 5 ]) . **The** hodograph circular character for **the** Kepler problem is closely related to **the** exis- tence **of** a speci ﬁ cally Keplerian constant **of** motion which which is an exceptional property **of** **the** central potential with radial dependence 1 r . From a purely historic viewpoint, this vector can be traced back to **the** beginning **of** **of** **the** 18th century, with Hermann **and** Bernoulli ( see two notes by Goldstein [ 6, 7 ]) , **and** was later rediscovered independently several times. **The** connection with **the** circular character **of** **the** hodograph seems to be due to Hamilton [ 1 ] ; from a modern viewpoint all these distinguished **properties** are linked to **the** superintegrability **of** **the** Kepler problem ( for a moderately advanced discussion, see [ 8 ]) .

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In **the** context **of** figural PG, Radford’s initial layer **of** structural generaliza- tion is factual, that is, it is “a **generalization** **of** numerical actions in **the** form **of** an operational scheme that remains bound to **the** numerical level, nevertheless allowing **the** students to virtually tackle any particular case successfully” (Rad- ford, 2001b, pp. 82-83). So, for example, one group **of** Grade 8 students in his study noticed that since **the** first two stages in pattern 1 **of** Figure 16 seem to fol- low **the** sense “it’s always **the** next… 1 + 2, 2 + 3,” that allowed them to impose **the** factual structure **of** “25 plus 26” in **the** case **of** Stage 25 **of** **the** pattern. Here **the** multiplicative dimension pertains to **the** two growing composite parts corre- sponding to **the** top **and** bottom rows **of** circles (versus **the** additive strategy **of** counting-all in which case circles are counted one by one **and** from stage to stage). Factual generalizations are often accompanied by **the** use **of** adverbs such as “**the** next” or “always,” including **the** effects **of** rhythm **of** an utterance **and** movement (e.g., a pointing gesture). While perhaps necessary in **the** beginning stage **of** generalizing, unfortunately, factual generalizations remain context- bound **and** numerical **and** often draw on shared “implicit agreements **and** mutual comprehension” (Radford, 2001b, p. 83) among those who construct them in so- cial activity.

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