In this paper we introduce the New/Generalized two parameter Gamma function and Pochhammer symbol. We named them, as Generalized p - k Gamma Function and Generalized p - k Pochhammer symbol and denoted as $ _{p}^{a}\Gamma_{k}(x) $ and $ _{p}^{a}(x)_{n,k} $ respectively. We prove the several identities for $ _{p}^{a}\Gamma_{k}(x) $ and $ _{p}^{a}(x)_{n,k} $ those satisfied by the classical Gamma function. Also we provide the integral representation for the $ _{p}^{a}\Gamma_{k}(x) $

In this note we construct explicit complex and real matrix representations for the generators of real Clifford algebra $C\ell_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. We find two classes of representation, the normal representation and exceptional one. The normal representation is a large class of representation which can only be expanded into $4m+1$ dimension, but the exceptional representation can be expanded as generators of the next period. In the cases $p+q=4m$, the representation is unique in equivalent sense. These results are helpful for both theoretical analysis and practical calculation. The generators of Clifford algebra are the faithful basis of $p+q$ dimensional Minkowski space-time or Riemann space, and Clifford algebra converts the complicated relations in geometry into simple and concise algebraic operations, so the Riemann geometry expressed in Clifford algebra will be much simple and clear.

In this paper, we provide some enlightening examples of the application of Clifford algebra in geometry, which show the concise representation, simple calculation and profound insight of this algebra. The definition of Clifford algebra implies geometric concepts such as vector, length, angle, area and volume, and unifies the calculus of scalar, spinor, vector and tensor, so that it is able to naturally describe all variables and calculus in geometry and physics. Clifford algebra unifies and generalizes real number, complex, quaternion and vector algebra, converts complicated relations and operations into intuitive matrix algebra independent of coordinate systems. By localizing the basis or frame of space-time and introducing differential and connection operators, Clifford algebra also contains Riemann geometry. Clifford algebra provides a unified, standard, elegant and open language and tools for numerous complicated mathematical and physical theories. Clifford algebra calculus is an arithmetic-like operation that can be well understood by everyone. This feature is very useful for teaching purposes, and popularizing Clifford algebra in high schools and universities will greatly improve the efficiency of students to learn fundamental knowledge of mathematics and physics. So Clifford algebra can be expected to complete a new big synthesis of scientific knowledge.

Succinic semialdehyde dehydrogenase deficiency (SSADH-D) is a genetic disorder that results from the aberrant metabolism of the neurotransmitter γ-amino butyric acid (GABA). The disease is caused by the impaired activity of the mitochondrial enzyme succinic semialdehyde dehydrogenase. SSADH-D manifests as varying degree of mental retardation, autism, ataxia and epileptic seizures, but the clinical picture is highly heterogeneous. So far, there is no approved therapy for this disease. In this review, we briefly summarize the molecular genetics of SSADH-D, the past and ongoing clinical trials and the emerging features of the molecular pathogenesis, including redox imbalance and mitochondrial dysfunction. The main aim of this review is to discuss the potential use of further therapy approaches that have so far not been tested in SSADH-D, such as pharmacological chaperones, read-through drugs and gene therapy. Special attention will also be paid to elucidating the role of patient advocacy organizations in facilitating research and in the communication between the researchers and the patients.

With new discoveries and insights in atomic and optical physics, the field of spectroscopy is advancing to a new level with applications ranging from material science to astronomy. We here propose a theoretical description of a potential new phenomenon resulting from the extension of Compton's experiment on the scattering of high energy photons through atomic electrons. How the proposed phenomenon can be tested experimentally is discussed in the paper as well.

In this paper, we derive multifarious relationships among the two types of higher order q-Daehee polynomials and p-adic gamma function via Mahler theorem. Also, we compute some weighted p-adic q-integrals of the derivative of p-adic gamma function related to the Stirling numbers of the both kinds and the q-Bernoulli polynomials of order k.

The main aim of this paper is to investigate multifarious properties and relations for the gamma distribution. The approach to reach this purpose will be introducing a special polynomial including gamma distribution. Several formulas covering addition formula, derivative property, integral representation and explicit formula are derived by means of the series manipulation method. Furthermore, two correlations including Bernoulli and Euler polynomials for gamma distribution polynomials are provided by utilizing of their generating functions.

In this paper, we introduce the (ρ,q)-analogue of the p-adic factorial function. By utilizing some properties of (ρ,q)-numbers, we obtain several new and interesting identities and formulas. We then construct the p-adic (ρ,q)-gamma function by means of the mentioned factorial function. We investigate several properties and relationships belonging to the foregoing gamma function, some of which are given for the case p = 2. We also derive more representations of the p-adic (ρ,q)-gamma function in general case. Moreover, we consider the p-adic (ρ,q)-Euler constant derived from the derivation of p-adic (ρ,q)-gamma function at x = 1. Furthermore, we provide a limit representation of aforementioned Euler constant based on (ρ,q)-numbers. Finally, we consider (ρ,q)-extension of the p-adic beta function via the p-adic (ρ,q)-gamma function and we then investigate various formulas and identities.

In this paper, we investigate several relations for p-adic gamma function by means of their Mahler expansion and fermionic p-adic q-integral on ℤ_{p}. We also derive two fermionic p-adic q-integrals of p-adic gamma function in terms of q-Boole polynomials and numbers. Moreover, we discover fermionic p-adic q-integral of the derivative of p-adic gamma function. We acquire a representation for the p-adic Euler constant by means of the q-Boole polynomials. We finally develop a novel, explicit and interesting representation for the p-adic Euler constant including Stirling numbers of the first kind.

The main aim of this paper is to set some correlations between Boole polynomials and p-adic gamma function in conjunction with p-adic Euler contant. We develop diverse formulas for p-adic gamma function by means of their Mahler expansion and fermionic p-adic integral on ℤ_{p}. Also, we acquire two fermionic p-adic integrals of p-adic gamma function in terms of Boole numbers and polynomials. We then provide fermionic p-adic integral of the derivative of p-adic gamma function and a representation for the p-adic Euler constant by means of the Boole polynomials. Furthermore, we investigate an explicit representation for the aforesaid constant covering Stirling numbers of the first kind.

In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on ℤ_{p} of p-adic gamma function via their Mahler expansions. We also derived two q-Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the first kind. We finally develop a novel and interesting representation for the p-adic Euler constant by means of the q-Daehee polynomials and numbers.

This paper establishes a real integral representation of the reciprocal $\Gamma$ function in terms of a regularized hypersingular integral. The equivalence with the usual complex representation is demonstrated. A regularized complex representation along the usual Hankel path is derived.

The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions are given and the conditions under which these function can be represented by simpler functions are demonstrated. The connection with generalized fractional differential and integral operators is demonstrated and discussed.

Chlorotoxin (CTX) is a 36–amino acid peptide with 8 Cys residues that forms four Cys-Cys bonds. It has high affinity for the glioma-specific chloride channel and matrix metalloprotease-2. Structural and binding properties of CTX analogs with various Cys residue substitutions with L-a-aminobutyric acid (Abu) have been previously reported. Using 4.2 ìs molecular dynamics, we compared the conformational and essential space sampling of CTX and analogs [Abu/Ser^2,19]CTX, [Abu/Ser^5,28]CTX, [Abu/Ser^16,33]CTX, [Abu/Ser^20,35]CTX, and [Abu/Ser^{2,5,16,19,20,28,33,35}]CTX. The native and substituted peptides maintained a high degree of a-helix propensity from residues 8 through 21, with the exception of [Ser^5,28]CTX and [Abu^16,33]CTX. In agreement with previous circular dichroism spectropolarimetry results, the C-terminal b-sheet content varied less from residues 25 through 29 and 32 through 36 and was well conserved in all analogs, except: [Abu^16,33]CTX, [Ser^20,35]CTX, [Abu^{2,5,16,19,20,28,33,35}]CTX, and [Ser^{2,5,16,19,20,28,33,35}]CTX. The Cys¹⁶-Cys³³ and Cys²⁰-Cys³⁵ Cys-Cys bonds appear to be required to maintain the ab-motif of CTX. Their selective substitution with Ser^16,33 however, may mitigate the destabilizing effect of Cys¹⁶-Cys³³ substitution by the formation of an inter residue H-bond from OgH of Ser¹⁶ to OgH of Ser³³ bridged by a water molecule. All peptides shared considerable sampled conformational space, which explains the retained receptor binding of the nonnative analogs.

Understanding the spatially varying effects of demographic factors on the spatio-temporal variation of intestinal parasites infections is important for public health intervention and monitoring. This paper presents a hierarchical Bayesian spatially varying coefficient model to evaluate the effects demographic factors on intestinal parasites morbidities in Ghana. The modeling relied on morbidity data collected by the District Health Information Management Systems. We developed Poisson and Poisson-gamma spatially varying coefficient Models. We used the demographic factors, unsafe drinking water, unsafe toilet and unsafe liquid waste disposal as model covariates. The models were fitted using the Integrated Nested Laplace Approximations (INLA). The overall risk of intestinal parasites infection was estimated to be 10.9 per 100 people with a wide spatial variation in the district-specific posterior risk estimates. Substantial spatial variation of increasing multiplicative effects of unsafe drinking water, unsafe toilet and unsafe liquid waste disposal occurs on the variation of intestinal parasites risk. The structured residual spatial variation widely dominates the unstructured component, suggesting that the unaccounted-for risk factors are spatially continuous in nature. The study concludes that both the spatial distribution of the posterior risk and the associated exceedance probability maps are essential for monitoring and control of intestinal parasites.

Over the past decade, our knowledge of the $\gamma$-ray sky has been revolutionized by ground- and space-based observatories by detecting photons up to several hundreds of tera-electron volt (TeV) energies. A major population of the $\gamma$-ray bright objects are active galactic nuclei (AGN) with their relativistic jets pointed along our line-of-sight. Gamma-ray emission is also detected from nearby mis-aligned AGN such as radio galaxies. While the TeV-detected radio galaxies ($TeVRad$) only form a small fraction of the $\gamma$-ray detected AGN, their multi-wavelength study offers a unique opportunity to probe and pinpoint the high-energy emission processes and sites. Even in the absence of substantial Doppler beaming $TeVRad$ are extremely bright objects in the TeV sky (luminosities detected up to $10^{45}~erg~s^{-1}$), and exhibit flux variations on timescales shorter than the event-horizon scales (flux doubling timescale less than 5 minutes). Thanks to the recent advancement in the imaging capabilities of high-resolution radio interferometry (millimeter very long baseline interferometry, mm-VLBI), one can probe the scales down to less than 10 gravitational radii in $TeVRad$, making it possible not only to test jet launching models but also to pinpoint the high-energy emission sites and to unravel the emission mechanisms. This review provides an overview of the high-energy observations of $TeVRad$ with a focus on the emitting sites and radiation processes. Some recent approaches in simulations are also sketched. Observations by the near-future facilities like Cherenkov Telescope Array, short millimeter-VLBI, and high-energy polarimetry instruments will be crucial for discriminating the competing high-energy emission models.

Gamma titanium aluminides (γ-TiAl) present an excellent behaviour under high temperature conditions, being a feasible alternative to nickel-based superalloy components in aeroengine sector. However, considered as a difficult to cut material, process cutting parameters require special study to guarantee components quality. In this work, developed drilling mechanistic model is a useful tool in order to predict drilling force (F_z) and torque (T_c) for optimal drilling conditions determination. The model is validated for three types of Gamma-TiAl alloys. Integral hard metal end-drilling tools and different cutting parameters (feeds and cutting speeds) are tested in three different sized holes for each alloy.

Kim-Kim ([9]) studied some properties of the degenerate gamma and degenerate Laplace transformation and obtained their properties. In this paper, we define modified degenerate gamma and modified degenerate Laplace Transformation and investigate some properties and formulas related to them.

Theories of quantum gravity suggest that Lorentz invariance, the fundamental symmetry of the Theory of Relativity, may be broken at the Planck energy scale. While any deviation from conventional Physics must be minuscule in particular at attainable energies, this hypothesis motivates ever more sensitive tests of Lorentz symmetry. In the photon sector, astrophysical observations, in particular polarization measurements, are a very powerful tool because tiny deviations from Lorentz invariance will accumulate as photons propagate over cosmological distances. The Standard-Model Extension (SME) provides a theoretical framework in the form of an effective field theory that describes low-energy effects due to a more fundamental quantum gravity theory by adding additional terms to the Standard Model Lagrangian. These terms can be ordered by the mass dimension d of the corresponding operator and lead to a wavelength, polarization, and direction dependent phase velocity of light. In this paper, we analyze optical polarization measurements from 63 Active Galactic Nuclei (AGN) and Gamma-ray Bursts (GRBs) in order to search for Lorentz violating signals. We derive the first set limits on each of the 10 individual birefringent coefficients of the minimal SME with d = 4, with 95% confidence limits on the order of 10⁻³⁴ on the dimensionless coefficients.

Metal, primarily Fe and Zn, deficiencies affect over half of the world's population. Human diets with prevalent cereal products cause micronutrient malnutrition. Biofortification is one of the most effective approaches to alleviate malnutrition. Spring wheat genetically stable (M7) mutant lines developed with 100 and 200 Gy gamma treatments to broaden genetic variation and search for new resources were analyzed for nutritionally important minerals (Ca, Mg, K, Fe, and Zn), their bioavailability, and grain protein content (GPC). The variation was 172.3–883.0 mg/kg for Ca, 472.9–1088 mg/kg for Mg, 3128.6–5487.5 mg/kg for K, 40.9–89.0 mg/kg for Fe, and 22.2–89.6 mg/kg for Zn. In mutant lines, among the investigated minerals, the highest increases in concentrations were observed in Fe, Zn, and Ca when compared to the parent. Some mutant lines, mostly in the 100 Gy-derived germplasm, had two to three times higher Fe, Zn, and Ca concentrations, lower phytic acid concentration (1.4–2.1 times), and 6.5–7% higher GPC compare to the parent. Variation was detected for the Ca:Phy, Mg :Phy, Phy:K, Phy:Fe, and Phy:Zn molar ratios, (1.27–10.41, 5.05–18.68, 1.66–4.87, 1.40–5.32 and 1.78–11.78, respectively). The results showed how the genetic variation could be generated through radiation and be useful to develop biofortification by micronutrient varieties with their appropriate bioavailability to overcome malnutrition.

The aim of this article is to establish a new class of unified integrals associated with the generalized Lommel-Wright functions, which are expressed in terms of Wright Hypergeometric function.Some integrals involving trigonometric,generalized Bessel function and Struve functions are also indicated as special cases of our main results.further, with the help of our main results and their special cases, we obtain two reduction formulas for the Wright hypergeometric function.

In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on Z_p of p-adic gamma function via their Mahler expansions. We also derived two q-Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the first kind. We finally develop a novel and interesting representation for the p-adic Euler constant by means of the q-Daehee polynomials and numbers.

Philippines as an archipelago and tropical country, which is situated near the Pacific ocean, faces uncertain rainfall intensities. This makes environmental, agricultural and economic systems affected by precipitation difficult to manage. Time series analysis of Philippine rainfall pattern has been previously done, but there is no study investigating its probability distribution. Modeling the Philippine rainfall using probability distributions is essential, especially in managing risks and designing insurance products. Here, daily and cumulative rainfall data (January 1961 - August 2016) from 28 PAGASA weather stations are fitted to probability distributions. Moreover, the fitted distributions are examined for invariance under subsets of the rainfall data set. We observe that the Gamma distribution is a suitable fit for the daily up to the ten-day cumulative rainfall data. Our results can be used in agriculture, especially in forecasting claims in weather index-based insurance.

Beginning in 2012, our understanding of human papillomavirus (HPV) subcellular trafficking has undergone a drastic paradigm shift. Work from multiple laboratories has revealed that HPV has evolved a unique means to deliver its viral genome (vDNA) to the cell nucleus, relying on a myriad of host cell proteins and processes. The major breakthrough finding from these recent endeavors was the realization of L2-dependent utilization of cellular sorting factors for the retrograde transport of vDNA away from degradative endo/lysosomal compartments to the Golgi, prior to mitosis-dependent nuclear accumulation of L2/vDNA. An overview of current models of HPV entry, subcellular trafficking, and the role of L2 during initial infection is provided below, highlighting unresolved questions and gaps in knowledge.

Many systems in nature and laboratories are far from equilibrium and exhibit significant fluc- tuations, invalidating the key assumptions of small fluctuations and short memory time in or near equilibrium. A full knowledge of Probability Distribution Functions (PDFs), especially time- dependent PDFs, becomes essential in understanding far-from-equilibrium processes. We consider a stochastic logistic model with multiplicative noise, which has gamma distributions as stationary PDFs. We numerically solve the transient relaxation problem, and show that as the strength of the stochastic noise increases the time-dependent PDFs increasingly deviate from gamma distributions. For sufficiently strong noise a transition occurs whereby the PDF never reaches a stationary state, but instead forms a peak that becomes ever more narrowly concentrated at the origin. The addition of an arbitrarily small amount of additive noise regularizes these solutions, and re-establishes the existence of stationary solutions. In addition to diagnostic quantities such as mean value, standard deviation, skewness and kurtosis, the transitions between different solutions are analyzed in terms of entropy and information length, the total number of statistically distinguishable states that a system passes through in time.

This article shortly introduces Gaussian processes as a new approach for modelling time series in the field of blazar physics. In the second part of the paper, recent results from an application of GP modelling to the multi-wavelength light curves of the blazar PKS 1502+106 are discussed.

In this paper, some convexity properties and some inequalities for the (p,k)-analogue of the Gamma function, Г_p,k(x) are given. In particular, a (p,k)-analogue of the celebrated Bohr-Mollerup theorem is given. Furthermore, a (p,k)-analogue of the Riemann zeta function, ζ_p,k(x) is introduced and some associated inequalities are derived. The established results provide the (p,k)-generalizations of some known results concerning the classical Gamma function.

In this paper, we establish some inequalities bounding the ratio Г_p,k(x)/Г_p,k(y), where Г_p,k(.) is the (p,k)-analogue of the Gamma function. Consequently, some previous results are recovered from the obtained results.

In a recent paper, Sarikaya et al. introduced a new analogue of the classical Gamma function which they called, the conformable Gamma function. Motivated by their results, this paper seeks to establish some inequalities for the conformable Gamma and Polygamma functions. Among other analytical tools, the procedure relies on the generalized forms of some classical inequalities.

In the paper, the authors generalize the notion "k-Mittag-Leffler function'', establish some integral transforms of the generalized k-Mittag-Leffler function, and derive several special and known conclusions in terms of the generalized Wright function and the generalized k-Wright function.

S5 0716+714 is a well known BL-Lac object, one of the brightest and most active blazars. The discovery in the Very High Energy band (VHE, E > 100 GeV) by MAGIC happened in 2008. In January 2015 the source went through the brightest optical state ever observed, triggering MAGIC follow-up and a VHE detection with ~ sigma significance (ATel #6999). Rich multi-wavelength coverage of the flare allowed us to construct the broad-band spectral energy distribution of S5 0716+714 during its brightest outburst. In this work we will present the preliminary analysis of MAGIC and Fermi-LAT data of the flaring activity in January and February 2015 for the HE and VHE band, together with radio (Metsahovi, OVRO, VLBA, Effelsberg), sub-millimeter (SMA), optical (Tuorla, Perkins, Steward, AZT-8+ST7, LX-200, Kanata), X-ray and UV (Swift-XRT and UVOT), in the same time-window and discuss the time variability of the MWL light curves during this impressive outburst.

The discovery by the Large Area Telescope on board Fermi of variable gamma-ray emission from radio-loud narrow-line Seyfert 1 (NLSy1) galaxies revealed the presence of a possible third class of AGN with relativistic jets in addition to blazars and radio galaxies. Considering that NLSy1 are usually hosted in spiral galaxies, this finding poses intriguing questions about the nature of these objects and the formation of relativistic jets. We report on a systematic investigation of the gamma-ray properties of a sample of radio-loud NLSy1, including the detection of new objects, using 7 years of Fermi-LAT data with the new Pass 8 event-level analysis. In addition we discuss the radio-to-very-high-energy properties of the gamma-ray emitting NLSy1, their host galaxy, and black hole mass in the context of the blazar scenario and the unification of relativistic jets at different scales.

High redshift blazars are among the most powerful objects in the Universe. Although they represent a significant fraction of the extragalactic hard X-ray sky, they are not commonly detected in gamma-rays. High redshift (z>2) objects represent <10 per cent of the AGN population observed by Fermi so far, and gamma-ray flaring activity from these sources is even more uncommon. The characterization of the radio-to-gamma-ray properties of high redshift blazars represents a powerful tool for the study of the energetics of such extreme objects and the Extragalactic Background Light. This contribution will present results of multi-band campaigns, from radio to gamma-rays, on PKS 0836+710, PKS 2149-306, and TXS 0536+145. The latter is the highest redshift detection of a flaring gamma-ray blazar so far. At the peaks of their respective flares these sources reached an apparent isotropic gamma-ray luminosity of about 10^50 erg/s, which is comparable with the luminosity observed from the most powerful blazars. The physical properties derived from the multi-wavelength observations of these sources are then compared with those shown by the high redshift population.

Radio loud active galactic nuclei are composed of different spatial features, each one characterized by different spectral properties in the radio band. Among them, blazars are the most common class of sources detected at gamma-rays by \fermi, and their radio emission is dominated by the flat spectrum compact core. In this contribution, we explore the connection between emission at high energy revealed by \fermi\ and at radio frequency. Taking as a reference the strong and very highly significant correlation found between gamma rays and cm-$\lambda$ radio emission, we explore different behaviours found as we change the energy range in gamma rays and in radio, therefore changing the physical parameters of the zones involved in the radiation emission. We find that the correlation weakens when we consider (1) gamma rays of energy above 10 GeV (except for high synchrotron peaked blazars) or (2) low frequency radio data taken by the Murchison Widefield Array; on the other hand, the correlation strengthens when we consider mm-$\lambda$ data taken by ALMA.

QSO B0218+357 is a blazar located at a cosmological redshift of 0.944. The source is gravitationally lensed by a spiral galaxy located at the redshift of 0.68. Strong gravitational lensing splits the signals emitted by the source into two components separated by 10-12 days, as observed in radio and gamma-ray bands. In July 2014 a GeV flare was observed by Fermi-LAT, triggering follow-up observations with the MAGIC telescopes at energies above 100 GeV. The expected time delay between the components allowed us to plan broad band MWL observations before, during and after the trailing component of the emission. The MAGIC observations at the expected time of arrival of the trailing component resulted in the first detection of QSO B0218+357 in Very-High-Energy (VHE, >100 GeV) gamma rays. It is both the farthest known VHE object and one of only a few Flat Spectrum Radio Quasars detected in this energy range. We report here the observed multiwavelength spectral and temporal properties of the emission during the 2014 flare.