On Fourier integral transforms for qq-Fibonacci and qq-Lucas polynomials

We study in detail two families of $q$-Fibonacci polynomials and $q$-Lucas polynomials, which are defined by non-conventional three-term recurrences. They were recently introduced by Cigler and have been then employed by Cigler and Zeng to construct novel $q$-extensions of classical Hermite polynomials. We show that both of these $q$-polynomial families exhibit simple transformation properties with respect to the classical Fourier integral transform

The National Dialogue on the Quadrennial Homeland Security Review

Six years after its creation, the Department of Homeland Security (DHS) undertook the first Quadrennial Homeland Security Review (QHSR) to inform the design and implementation of actions to ensure the safety of the United States and its citizens. This review, mandated by the Implementing the 9/11 Commission Recommendations Act of 2007, represents the first comprehensive examination of the homeland security strategy of the nation. The QHSR includes recommendations addressing the long-term strategy and priorities of the nation for homeland security and guidance on the programs, assets, capabilities, budget, policies, and authorities of the department.Rather than set policy internally and implement it in a top-down fashion, DHS undertook the QHSR in a new and innovative way by engaging tens of thousands of stakeholders and soliciting their ideas and comments at the outset of the process. Through a series of three-week-long, web-based discussions, stakeholders reviewed materials developed by DHS study groups, submitted and discussed their own ideas and priorities, and rated or "tagged" others' feedback to surface the most relevant ideas and important themes deserving further consideration.Key FindingsThe recommendations included: (1) DHS should enhance its capacity for coordinating stakeholder engagement and consultation efforts across its component agencies, (2) DHS and other agencies should create special procurement and contracting guidance for acquisitions that involve creating or hosting such web-based engagement platforms as the National Dialogue, and (3) DHS should begin future stakeholder engagements by crafting quantitative metrics or indicators to measure such outcomes as transparency, community-building, and capacity

The Reversed q-Exponential Functional Relation

After obtaining some useful identities, we prove an additional functional relation for $q$ exponentials with reversed order of multiplication, as well as the well known direct one in a completely rigorous manner.Comment: 6 pages, LaTeX, no figure

q-Analogue of Shock Soliton Solution

By using Jackson's q-exponential function we introduce the generating function, the recursive formulas and the second order q-differential equation for the q-Hermite polynomials. This allows us to solve the q-heat equation in terms of q-Kampe de Feriet polynomials with arbitrary N moving zeroes, and to find operator solution for the Initial Value Problem for the q-heat equation. By the q-analog of the Cole-Hopf transformation we construct the q-Burgers type nonlinear heat equation with quadratic dispersion and the cubic nonlinearity. In q -> 1 limit it reduces to the standard Burgers equation. Exact solutions for the q-Burgers equation in the form of moving poles, singular and regular q-shock soliton solutions are found.Comment: 13 pages, 5 figure

h analogue of Newton's binomial formula

In this letter, the $h$--analogue of Newton's binomial formula is obtained in the $h$--deformed quantum plane which does not have any $q$--analogue. For $h=0$, this is just the usual one as it should be. Furthermore, the binomial coefficients reduce to $\frac{n!}{(n-k)!}$ for $h=1$. \\ Some properties of the $h$--binomial coefficients are also given. \\ Finally, I hope that such results will contribute to an introduction of the $h$--analogue of the well--known functions, $h$--special functions and $h$--deformed analysis.Comment: 6 pages, latex Jounal-ref: J. Phys. A: Math. Gen. 31 (1998) L75

Central factorials under the Kontorovich-Lebedev transform of polynomials

We show that slight modifications of the Kontorovich-Lebedev transform lead to an automorphism of the vector space of polynomials. This circumstance along with the Mellin transformation property of the modified Bessel functions perform the passage of monomials to central factorial polynomials. A special attention is driven to the polynomial sequences whose KL-transform is the canonical sequence, which will be fully characterized. Finally, new identities between the central factorials and the Euler polynomials are found.Comment: also available at http://cmup.fc.up.pt/cmup/ since the 2nd August 201

Nanodiamond quantum sensors reveal temperature variation associated to hippocampal neurons firing

Temperature is one of the most relevant parameters for the regulation of intracellular processes. Measuring localized subcellular temperature gradients is fundamental for a deeper understanding of cell function, such as the genesis of action potentials, and cell metabolism. Here, we detect for the first time temperature variations (1{\deg}C) associated with potentiation and depletion of neuronal firing, exploiting a nanoscale thermometer based on optically detected magnetic resonance in nanodiamonds. Our results provide a tool for assessing neuronal spiking activity under physiological and pathological conditions and, conjugated with the high sensitivity of this technique (in perspective sensitive to < 0.1{\deg}C variations), pave the way to a systematic study of the generation of localized temperature gradients. Furthermore, they prompt further studies explaining in detail the physiological mechanism originating this effect.Comment: 27 pages, 5 figures, 3 table