Although most of these notes were written without consulting outside sources, there are a few exceptions which I should call attention to. The proof that complex lattices are classified by the fundamental domain is based on that in [4, Theorem VII.1]. The approach to the proof of unique factorization of ideals is inspired by that of [1, Section XI.8]. The material on Dirichletseries is a synthesis of that in [2, Lemma VII.1] and [3, Theorem 7.11] (rewritten to avoid any explicit discussion of absolute convergence). Finally, the proof of the key estimate on lattice points of bounded absolute value is taken from [2, pp. 160–161].
the contacts are considered point-like, which in practice means that their diameter w (the same is also true for c). The practical purpose of solving the boundary-value is to determine the voltage and thus the stability of the results with respect to the displacement and w . In , these results were compared to those associated with a non-rotated array. When s = 0 and w/a = 1/2, we obtain the situation shown in Figure 2, where all four contacts are on the boundary in a highly symmetric arrangement.
Fractional calculus is a very rapidly growing subject of mathematics which deals with the study of fractional order derivatives and integrals. Fractional calculus is an eﬃcient tool to study many complex real world systems . It is demonstrated that the fractional or- der representation of complex processes appearing in various ﬁelds of science, engineer- ing and ﬁnance, provides a more realistic approach with memory eﬀects to study these problems (see e.g. [2–13]). Among the research work developing the theory of fractional calculus and presenting some applications, we point out some literature. Kumar et al.  analyzed the fractional model of a modiﬁed Kawahara equation by using a newly intro- duced Caputo–Fabrizio fractional derivative. One also  studied a heat transfer prob- lem and presented a new non-integer model for convective straight ﬁns with temperature- dependent thermal conductivity associated with Caputo–Fabrizio fractional derivative. Recently, one  presented a new fractional extension of regularized long wave equation by using an Atangana–Baleano fractional operator. In  one introduced a new numeri- cal scheme for a fractional Fitzhugh–Nagumo equation arising in the transmission of new impulses. In  one constituted a modiﬁed numerical scheme to study fractional model of Lienard’s equations. Hajipour et al.  formulated a new scheme for a class of fractional chaotic systems. Baleanu et al.  proposed a new formulation of the fractional control problems involving a Mittag-Leﬄer non-singular kernel. In another work, Baleanu et al.  studied the motion of a bead sliding on a wire in a fractional analysis. Jajarmi et al.  analyzed a hyperchaotic ﬁnancial system and its chaos control and synchronization by using fractional calculus.
It is interesting to mention here that whenever a generalized hypergeometric function reduces to gamma functions, the results are very important from the application point of view. Thus well-known classical summation theorems such as those of Gauss, Gauss sec- ond, Kummer, and Bailey for the series F ; Watson, Dixon, Whipple and Saalschütz for
In ecology and epidemiology, distributions of events like disease occurrences, predator arrivals or plant locations can be considered as realizations of a point process, of which each point represents a single event. The point process theory has been presented and discussed in ,  and . Statistical procedures for analysing point process realizations can be found in books (; ; ; ) and a lot of papers deal with applications in special situa- tions (e.g. ; ; ; ). Some studies are based on counts of events in sampling units (), and some others on event spatial positions or occurrence dates (), and also distance sampling (). Perry et al. () discussed appropriate selection and use of method for analyzing spatial point patterns in plant ecology. One may refer to  for papers about statistical tools for spatial point processes.  discussed recently about spatial point process models for forest inventories exhibiting overdispersion. In applications in which overdispersion is assumed, Cox process modeling is a com- mon choice since this class of point processes is wide enough to take into consideration many features. Thus,  presented various scientific fields in which the Cox process, also known as doubly stochastic Poisson process, occurs. The intensity process λ(.) of
Next, the introduction of the Rodrigues formulas (4) and (5) leads us to generalization of many well-known Rodrigues formulas up to fractional forms. In this regard the Rodrigues rep- resentations (4) and (5), in particular, yield the following new fractional Rodrigues–type repre- sentations for the generalized Hermite polynomials H n (r) (x , a, b) , the generalized Laguerre polynomials L (α) n ( x , k, p) , Bessel polynomials y n ( x ) and Humbert polynomials h n (x ) as follows:
heterogeneous information modalities, i.e., the visual modality referring to low-level visual features and the semantic modality referring to high-level human concepts. To bridge the semantic gap, we present an extension of latent Dirichlet allocation (LDA), denoted as class-specific Gaussian-multinomial latent Dirichlet allocation (csGM-LDA), in an effort to simulate the human’s visual perception system. An analysis of previous supervised LDA models shows that the topics discovered by generative LDA models are driven by general image regularities rather than the semantic regularities for image annotation. To address this, csGM-LDA is introduced by using class supervision at the level of visual features for multimodal topic modeling. The csGM-LDA model combines the labeling strength of topic supervision with the flexibility of topic discovery, and the modeling problem can be effectively solved by a variational expectation-maximization (EM) algorithm. Moreover, as natural images usually generate an enormous size of
We then use the formula to derive the q-analogue of Gauss summation formula and to obtain a number of etafunction, q-gamma, and q-beta function identities, which complement the works of Bhargava and Somashekara 6, Bhargava et al. 7, Somashekara and Mamta 8, Srivastava 9, and Bhargava and Adiga 10.
The purpose of the present study was to further our understanding of the role series elastic compliance plays in manipulating force generation and explore the possibility that the extent of active shortening afforded by series compliance may affect force summation. We have shown that modest reductions in the amplitude and velocity of active shortening owing to an increase in effective SEE stiffness were sufficient to substantially augment the mechanical properties of the twitch. Twitch torque, TTI and peak RTD were considerably greater for contractions where active fascicle shortening was restricted by a rapid, small-amplitude stretch than for constant-length contractions performed at the initial and final MTU lengths of the rotation contraction. Our results also show that the torque contribution of a second stimulus was sensitive to the degree of active shortening permitted during the preceding period of contractile activity. The torque contribution increased when prior active shortening was restricted by a transient increase in effective SEE stiffness. As active fascicle lengths post-rotation were shorter than fascicle lengths measured during the constant-length contraction at the final MTU length, our findings suggest that history-dependent properties may influence the force-generating potential of stimuli within a burst.
I f = f x d x are carefully studied. This research is a continuation of the results in the -. All these quadrature formulas are not based on the inte- gration of an interpolant as so as the Gregory rule, a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order (see ). In some natural restric- tions on the parameters we construct the only one quadrature formula of the eight order which belongs to the first, second and third family. For functions whose 8th derivative is either always positive or always negative, we use these quadrature formulas to get good two-sided bound on
It is difficult to calculate complete General Formula for summation of n terms in such Pth Level Arithmetic series. However the sum of first three terms alone in abovesaid Pth level General formula gives an upper bound beyond which the sum of n-terms in a series, where each term is raised to power P, cannot exceed. The said General Formula , gives exact result for P<=3 where P>0 and is an upper bound for Series with P>3 , differing from the exact sum of the terms in the series by less than 0.01% , when the number of terms in series exceed or equal 2P, where P is the power to which the terms in Series are raised to. The said General Formula can therefore be limited to sum of first three terms and expressed as --
Under the PACE program, property owners subject to a PACE assessment make property tax payments to the County, which then remits all collections associated with the PACE assessments to the PACE Bond Trustee every January, May and August. Collections on PACE assessments are not separated from general tax collections until the delivery dates in January, May and August occur, with the effect that until the funds are remitted to the PACE Bond Trustee, PACE assessment collections are commingled with other revenues of the County and may be subject to an automatic stay in the event of a municipal bankruptcy of the County. KBRA considered the following mitigating factors in the analysis:
Recently good progress has been done in the direction of generalizing the above- mentioned classical summation theorems 2.2–2.7 see 6. In fact, in a series of three papers by Lavoie et al. 7–9, a large number of very interesting contiguous results of the above mentioned classical summation theorems 2.2–2.7 are given. In these papers, the authors have obtained explicit expressions of
asymptotic behavior of solutions of equation (.) under the Dirichlet boundary condition. In , Gourley considered the existence of travelling front solutions and their qualitative form for equation (.). The nonlinear stability of travelling wavefronts of equation (.) was investigated by Mei et al. in . Yi and Zou  also established the global attractivity of the positive steady state of equation (.). Yi et al. in  established the threshold dy- namics of equation (.) subject to the homogeneous Dirichlet boundary condition when the delayed reaction term is non-monotone.
 B. Brubaker, D. Bump, G. Chinta, S. Friedberg, J. Hoffstein, Weyl group multiple Dirichletseries, Pro- ceedings of the Workshop on Multiple DirichletSeries held in Bretton Woods, NH, July 11–14, 2005. Proceedings of Symposia in Pure Mathematics, 75. American Mathematical Society, Providence, RI, 2006.  V. Bykovskii, Functional equations for Hecke-Maass series, Funct. Anal. and Appl., 34 (2) (2000), 98–105.  N. Diamantis, D. Goldfeld, A converse theorem for double Dirichletseries. Amer. J. of Math., 133(4),
Looking for counterexamples to the Grand Riemann Hypothesis (GRH), some Dirichletseries satisfying a Riemann type of functional equation have been found, whose analytic continuation exhibit off critical line non trivial zeros, namely the Davenport and Heilbronn type of functions and linear combinations of L-functions satisfying the same functional equation. Although these are not counterexamples to GRH, their study allowed us to draw interesting conclusions. We have seen in  that if ζ A , Λ ( ) s does not satisfy the GRH, then for every