Search

A new semi-parametric Expected Shortfall (ES) estimation and forecasting framework is proposed. The proposed approach is based on a two-step estimation procedure. The first step involves the estimation of Value-at-Risk (VaR) at different quantile levels through a set of quantile time series regressions. Then, the ES is computed as a weighted average of the estimated quantiles. The quantiles weighting structure is parsimoniously parameterized by means of a Beta weight function whose coefficients are optimized by minimizing a joint VaR and ES loss function of the Fissler-Ziegel class. The properties of the proposed approach are first evaluated with an extensive simulation study using two data generating processes. Two forecasting studies with different out-of-sample sizes are then conducted, one of which focuses on the 2008 Global Financial Crisis (GFC) period. The proposed models are applied to 7 stock market indices and their forecasting performances are compared to those of a range of parametric, non-parametric and semi-parametric models, including GARCH, Conditional AutoRegressive Expectile (CARE), joint VaR and ES quantile regression models and simple average of quantiles. The results of the forecasting experiments provide clear evidence in support of proposed models., 38 pages, 2 figures and 6 tables

In this paper, we propose a numerical method for solving distributed-order fractional partial differential equations (FPDEs). For this method, we first introduce fractional-order generalized Taylor wavelets (FOGTW). An estimation for the error of the approximation is also studied. In addition, by using the regularized beta function we give a formula for determining the Riemann-Liouville fractional integral operator for the FOGTW. Combining this formula with the Gauss-Legendre quadrature, we obtain a numerical method for solving distributed-order FPDEs. Several illustrative examples are given to show the applicability and the accuracy of the proposed method.

Particle size control in mineral processing is typically achieved using an index characterising the whole distribution, e.g. the sieve opening through which a given mass fraction of the population would pass. Although mineral liberation is the goal of size reduction, its use for monitoring and process control applications is not reported because its online measuring is currently not possible. This paper proposes to address this issue using a mineral liberation distribution estimator consisting in two models. The first one uses a polynomial expression to relate the size-by-size mineralogical distribution of every mineral species to that of the ore. The second model relies on a beta distribution function to sort the masses in every size fraction into liberation classes. The characteristic constants of the beta function are parameterised with the properties of the ore. Results demonstrate that the liberation estimator can represent liberation distributions of both fresh and processed comminution products.

In this paper, we obtain a ( p , v ) -extension of the Appell hypergeometric function F 1 ( ⋅ ) , together with its integral representation, by using the extended Beta function B p , v ( x , y ) introduced in Parmar et al. (2015). Also, we give some of its main properties, namely the Mellin transform, a differential formula, recursion formulas and a bounded inequality. In addition, some new integral representations of the extended Appell function F 1 , p , v ( ⋅ ) involving Meijer’s G -function are obtained.

The huge complexity and uncertainty in real life requires the use of advanced automatic learning methods to find out better approximators and suitable relationship in real data behavior. Neuro fuzzy systems have been proved to be excellent universal approximators. In this paper we propose a new based function Interval Type-2 Fuzzy Neural Network denoted ”Beta basis function Interval Type-2 Fuzzy Neural Network”, the BIT2FNN. The main idea is to involve type-2 beta fuzzy sets in the design process of fuzzy networks. The proposed architecture is based on beta type-2 fuzzy sets in the antecedent part, while the consequent part achieves the TSK (Takagi–Sugeno–Kang) fuzzy output strategy. Thanks to the beta function flexibility, the network achieve a good performance and shows a good resistance to noisy data. First order derivatives of type-1 and type-2 Beta functions were developed for the first time for designing fuzzy logic systems based on given input–output pairs. The backpropagation algorithm was used for the learning process of antecedent fuzzy beta parameters and the consequent part. The performance of the proposed model of Beta fuzzy logic system is evaluated with mainly two problems of time series applications : the Mackey Glass Chaotic Time-Series prediction problem with different setting of parameters and levels of noise and the ECG heart-rate Time Series monitoring problem.

The scattering amplitudes of gauge bosons in heterotic and open superstring theories are related by the single-valued projection which yields heterotic amplitudes by selecting a subset of multiple zeta value coefficients in the $\alpha'$ (string tension parameter) expansion of open string amplitudes. In the present work, we argue that this relation holds also at the level of low-energy expansions (or individual Feynman diagrams) of the respective effective actions, by investigating the beta functions of two-dimensional sigma models describing world-sheets of open and heterotic strings. We analyze the sigma model Feynman diagrams generating identical effective action terms in both theories and show that the heterotic coefficients are given by the single-valued projection of the open ones. The single-valued projection appears as a result of summing over all radial orderings of heterotic vertices on the complex plane representing string world-sheet., Comment: 28 pages

Let Γ 3 be the triple gamma function. Our main objective is to prove that the function log ⁡ Γ 3 ( z + 1 ) − Γ 3 ′ ( 1 ) z z 3 Log z + Γ 3 ′ ( 1 ) z − 1 , z ∈ C ∖ ( − ∞ , 0 ] is a Pick function and find its Stieltjes representation.

Based on explorations in estimating certain Madelung constants, we put forward here two separate strategies to understand the meaning of two distinct classes of divergent non-power-series expansions. One class refers to alternating series representations, the other to monotonic ones. They chiefly rest on precise and approximate polynomial extrapolations, depending on situations. In case of sawtooth sequences, e.g., the partial-sums obtainable from Dirichlet eta or beta function at negative integer arguments, exact sequence-generating polynomials are found. Extrapolations yield a graphical meaning to anti-limit here, along with the exact answer. For staircase sequences, like the ones obtained from partial-sums of series representations for lambda and zeta functions, again at negative integer arguments, anti-limits do not exist. But, correct sequence-generating polynomials are obtained. There, our recipe relies on estimation of specific, finite areas embedded by such polynomials. The schemes put forward here are direct, independent and conceptually appealing. A subsequent extension of the latter strategy to alternating series also lends extra credence. Two new interpretations of summability are gained. Pilot calculations on several types of lattice sums reveal the worth of our endeavor with approximate extrapolations as well.

In this paper, the following multiple fractional part integrals I n , β α 1 , α 2 , ? , α n = ? 0 , 1 n ? j = 1 n x j α j { S n - 1 } β d x 1 ? d x n and J n , β α = ? 0 , 1 n S n α { S n - 1 } β d x 1 ? d x n are studied for positive integer n and complex values of α , β , α j ( j = 1 , 2 , ? , n ) , where { u } denotes the fractional part of u , R ( s ) denotes the real part of s and S n = x 1 + x 2 + ? + x n . It is proved that I 1 , β α can be represented as a linear combination of the Riemann zeta function, the Beta function and Euler's constant as R ( β ) - 1 . Moreover, I n , β α 1 , α 2 , ? , α n can be expressed by I n - 1 , β α 1 , α 2 , ? , α n - 1 , the Beta function and the incomplete Beta function for n = 2 , 3 . In addition, the recurrence formula of J n , β α ( n = 2 , 3 , ? ) is established and J n , β α can be expressed by I 1 , β α , logarithmic function and some binomial coefficients.

Considering different parameters and by means of Hadamard’s inequality, we obtain new and more general half-discrete Hilbert-type inequalities. Then we extract from our results some special cases that have been proved previously by other authors.

Solving a linear partial differential equation witness a noteworthy role of Wright function. Due to its usefulness and various applications, a variety of its extensions (and generalizations) have been investigated and presented. The purpose and design of the paper are intended to study and come up with a new extension of the generalized Wright function by using generalized beta function and obtain some integral representation of the freshly defined function. Also, we present the Mellin transform of this function in the form of Fox–Wright function. Furthermore, we obtain the recurrence relation, derivative formula for the said function and also by using an extended Riemann–Liouville fractional derivative, we present a fractional derivative formula for the generalized Wright function.

In this work, a q -analogue of the Gauss product formula for the q -Gamma function Γ q and a q -analogue of Euler's formula for the q -Beta function B q are established.

A new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel and parameters is given. The constant factor related to the hypergeometric function and the beta function is proved to be the best possible. As applications, equivalent forms, the reverses, some particular examples, two kinds of Hardy-type inequalities, and operator expressions are considered.

Open string disk integrals are represented as contour integrals of a product of Beta functions using Mellin transformations. This makes the mathematical problem of computing the α ′ -expansion around the field-theory limit similar to that of the ϵ -expansion in Feynman loop integrals around the four-dimensional limit. More explicitly, the formula in Mellin space obtained directly from the standard Koba–Nielsen-like representation is valid in a region of values of α ′ that does not include α ′ = 0 . Analytic continuation is therefore needed since contours are pinched by poles as α ′ → 0 . Deforming contours that get pinched by poles generates a set of ( n − 3 ) ! multi-dimensional residues left behind which contain all the field theory information. Some analogies between the field theory formulas obtained by this method and those derived recently from using the scattering equations are commented at the end.

In this paper, we establish some new refinements and similar extensions of Hermite-Hadamard inequality for fractional integrals and present several applications in the Beta function.

In this paper a multidimensional Hilbert-type integral inequality with the best possible constant factor associated to the gamma function is proved. An equivalent form and some reverses are obtained. We also consider the operator expressions and some particular results related to the non-homogeneous kernels and a homogeneous kernel of degree zero.

In this article, we present some completely monotone properties for an arbitrary real parameter function involving the Gamma function. As an application of these results, some meaningful and useful inequalities are obtained.

We calculated the one-loop contribution to the Callan–Symanzik beta function which is induced by the non-Abelian tensor gauge fields. The contribution is negative and corresponds to the asymptotically free theory. Two methods have been used to calculate the beta function, the first one based on generalization of the Altarelli–Parisi momentum sum rule, which implicitly comprises unitarity, and the second one – on the effective Lagrangian approach. Possible consequences are discussed.

The gamma and beta functions have been generalized in several ways. The multivariate beta and multivariate gamma functions due to Ingham and Siegel have been defined as integrals having the integrand as a scalar function of the real symmetric matrix. In this article, we define extended matrix variate gamma and extended matrix variate beta functions thereby generalizing multivariate gamma and multivariate beta functions defined by Ingham and Siegel. We study a number of properties of these newly defined functions. We also give some applications of these functions to statistical distribution theory.

We present the results for three-loop beta-function for the Higgs self-coupling calculated within the unbroken phase of the Standard Model. We also provide the expression for three-loop beta-function of the Higgs mass parameter, which is obtained as a by-product of our main calculation. Our results coincide with that of recent paper arXiv:1303.2890. In addition, the expression for the Higgs field anomalous dimension is given., 15 pages, 1 figure, Mathematica outputs attached. Ancillary files rearranged. arXiv admin note: text overlap with arXiv:1212.6829

In this paper we provide an application of the Euler-Maclaurin summation formula with the Bernoulli function for the proof of a strengthened version of the half-discrete Hilbert inequality with the best constant factor in terms of the Euler-Mascheroni constant. Some equivalent numerical representations, operator representations, two kinds of reverses as well as an extension in terms of parameters and the Beta function are also studied.

We present the results for three-loop beta-functions for Yukawa couplings of heavy Standard Model fermions calculated within the unbroken phase of the model. The calculation is carried out with the help of the MINCER program in a general linear gauge, and the final result is independent of the gauge-fixing parameters. In order to calculate three-point functions, we made use of infrared rearrangement (IRR) trick. Due to the chiral structure of the SM a careful treatment of loops with fermions is required to perform the calculation. It turns out that gauge anomaly cancellation in the SM allows us to obtain the result by means of the semi-naive treatment of gamma5., Comment: 7 pages, 1 figure, discussion about gamma5 extended, numerical estimates changed according to new ancillary files with correct results

Stirling’s expansions for Γ ( z ) and log Γ ( z ) for large z are well known. Here we give expansions for the Beta function B ( a , z + b ) and its reciprocal for large z and fixed ( a , b ) .

This work investigates the impact of global climate change on the sustainable growth of forest, namely, on its aggregated characteristics such as the number of trees, the basal area, and the amount of carbon sequestrated in the stand. The forest dynamics is described by a nonlinear size-structured population model. The existence of a steady state regime is proven and explicit formulas for the aggregated characteristics are obtained. A numeric simulation on realistic data illustrates and extends the analytic results obtained The work was supported by the NSF DMS grant 1009197 and the Ministerio de Ciencia e Innovacion, ECO2010-17020, Generalitat de Catalunya (XREPP, 2009 SGR189)

In this paper we establish a general method for improving the discrete Hilbert-type inequalities via the Hermite–Hadamard inequality. The general result is then applied to homogeneous kernels. Moreover, some particular homogeneous kernels are considered, and a whole series of refinements of some recent results known from the literature are yielded. In addition some particular non-homogeneous cases are also considered.

For N = 1 supersymmetric quantum electrodynamics, regularized by higher derivatives, a method for summation of all Feynman diagrams defining the β-function is presented. Using this method we prove that the β-function is given by an integral of a total derivative, which can be easily calculated. It is shown that surviving terms give the exact NSVZ β-function. The results are compared with the explicit three-loop calculation.

There is an abundant literature on inequalities for the Gamma function Γ and its various related functions as well as their approximations. Only very recently, several authors began to investigate various inequalities for the double Gamma function Γ 2 and its approximation. Here, in this sequel to some of these recent works, we aim at presenting an integral representation of the triple Gamma function Γ 3 , which is then used to derive an asymptotic formula for Γ 3 . As a by-product of the results presented here, integral representations and asymptotic formulas for the Gamma function Γ and the double Gamma function Γ 2 are also given. The methods and techniques used in this paper can easily be extended to derive the corresponding integral representations and asymptotic formulas for the multiple Gamma functions Γ n ( n ≧ 4).

Low emittance is a primary goal of electron storage ring based light sources. Optics with low emittance always require low beta functions in dipoles, which implies that strong quadrupoles are needed and that natural chromaticity arises. The work presented in this paper demonstrates that by introducing dipoles with a longitudinal dipolar field-variation, the minimum emittance can be lowered even if the minimum beta function is constrained.

In this article, we derive several properties such as marginal distribution, moments involving zonal polynomials, and asymptotic expansion of the complex bimatrix variate beta type 1 distribution introduced by Di´az-Garci´a and Gutierrez Jaimez [Jose A. Di´az-Garci´a, Ramon Gutierrez Jaimez, Complex bimatrix variate generalised beta distributions, Linear Algebra Appl. 432 (2010) 571–582]. We also derive distributions of several matrix valued functions of random matrices jointly distributed as complex bimatrix variate beta type 1.

The infrared electric images always have poor vision effects due to both the limitation in thermograph technology and surrounding interferences. Thus, the enhancement of such images is of great significance. A fast self-adaptive enhancement algorithm is presented on the basis of Bate function. In order to meet the real-time demand of online monitoring, a fast algorithm is put forward to complete the histogram transform by table-look-up operations, and thus the time-consuming repeats of integration of Beta function is avoided. Furthermore, PSO algorithm is combined with Chaos system, and then is employed to search the parameters of Beta function adaptively. The experimental results validate that the present method is effective and practical.

We investigate several properties of the linear Aouf–Silverman–Srivastava operator and associated classes of multivalent analytic functions which were introduced and studied by Aouf et al. [M.K. Aouf, H. Silverman, H.M. Srivastava, Some families of linear operators associated with certain subclasses of multivalent functions, Comp. Math. Appl. 55 (2008) 535–549]. Several theorems are extensions of earlier results of the above paper.

A novel finite element topology optimization procedure is presented based on the application of probability density and cumulative distribution functions. The procedure utilizes a family of Beta functions with constant probability mean which provide a smooth transition from a uniform to a bi-modal density distribution while conserving constant mean density and therefore constant mass. Validation of the method is demonstrated for several well-known two-dimensional minimum-weight structures. A general minimum-weight cylindrical structural layout for the support of any combination of axial and torsional loading has been developed to provide a test case for three dimensional numerical topological optimization. It is observed that this solution presents a challenge, especially for cases where the axial load is significantly larger than the torsional loading. For these cases, slender members are an essential part of the optimal topology.

In this paper, we prove some generalisations of several theorems given in [K.A. Driver, S.J. Johnston, An integral representation of some hypergeometric functions, Electron. Trans. Numer. Anal. 25 (2006) 115–120] and examine some special cases which correspond to a transformation given by Chaundy in [T.W. Chaundy, An extension of hypergeometric functions, I., Quart. J. Maths. Oxford Ser. 14 (1943) 55–78] and other transformations involving the Riemann zeta function and the beta function.

In this paper we give a sufficient condition for function to be α-starlike function and some of its applications. We use the techniques of convolution and differential subordinations.

Using Wilsonian methods, we study the renormalization group flow of the Nonlinear Sigma Model in any dimension $d$, restricting our attention to terms with two derivatives. At one loop we always find a Ricci flow. When symmetries completely fix the internal metric, we compute the beta function of the single remaining coupling, without any further approximation. For $d>2$ and positive curvature, there is a nontrivial fixed point, which could be used to define an ultraviolet limit, in spite of the perturbative nonrenormalizability of the theory. Potential applications are briefly mentioned., Comment: 9 pages

This work presents a comprehensive approach to estimate kinetic parameters when the involved reactions contain lumped chemical species. This approach is based on representing rate constants with a continuous probability distribution function. In particular, the beta function is used to estimate kinetic parameters in catalytic cracking reactions. Thus, several kinetic models for the catalytic process containing different number of lumps are selected and a discretization procedure is carried out to estimate the corresponding kinetic parameters. The kinetic representation based on the probability distribution substantially reduces the computational and experimental effort involved in the numerical evaluation of kinetic parameters.

We consistently incorporate Yang Mills matter fields into string corrected (deformed), D=10, N=1 Supergravity. We solve the Bianchi identities within the framework of the modified beta function favored constraints to second order in the string slope parameter $\g$ also including the Yang Mills fields. In the torsion, curvature and H sectors we find that a consistent solution is readily obtained with a Yang Mills modified supercurrent $A_{abc}$. We find a solution in the F sector following our previously developed method., Comment: 11 pages, LaTeX file, PACS number: 04.65.+e

The incomplete beta function B x ( a , b ) is defined for a , b > 0 and 0 x 1 . Its definition can be extended, by regularization, to negative non-integer values of a and b. In this paper we define the incomplete beta function B x ( a , b ) for negative integer values of a and b. Further we prove that the function ∂ m + n ∂ a m ∂ b n B x ( a , b ) exists for m , n = 0 , 1 , 2 , … and all a and b.

We, by making use of elementary arguments, deduce integral representations of the Legendre chi function $\chi_{s}(x)$ valid for $|z|1$. Our earlier established results on the integral representations for the Riemann zeta function $\zeta(2 n+1)$ and the Dirichlet beta function $\beta(2 n)$ ,$ n\in\mathbb{N}$, are direct consequence of these representations., Comment: 13 pages

In this paper, the exact form of Fisher information matrix for a four-parameter kappa distribution (K4D) is determined. The K4D is so general that includes a variety of distributions as special cases. The necessary condition for the existence of Fisher information matrix is { 0 h 1 2 , k 1 2 } ∪ { h 0 , 1 2 h k 1 2 } for k ≠ 0 , h ≠ 0 .

In this paper, we obtain the evaluations of certain Euler type integrals. Further, we establish a theorem on extended beta function and apply this to obtain evaluations of some integrals in terms of extended beta function B ( α , β ; A ) . Furthermore, we derive a number of new results as applications of these evaluations. We extend also some results of this paper to the multi-variable case.

We consider the renormalisation group flow of gauge couplings within the so-called exceptional supersymmetric standard model (E$_6$SSM) based on the low energy matter content of 27 dimensional representations of the gauge group $E_6$, together with two additional non-Higgs doublets. The two--loop beta functions are computed, and the threshold corrections are studied in the E$_6$SSM. Our results show that gauge coupling unification in the E$_6$SSM can be achieved for phenomenologically acceptable values of $\alpha_3(M_Z)$, consistent with the central measured low energy value, unlike in the minimal supersymmetric standard model (MSSM) which, ignoring the effects of high energy threshold corrections, requires significantly higher values of $\alpha_3(M_Z)$, well above the experimentally measured central value., Comment: 15 pages, 1 figure, references added, section 4 expanded, table added

We present a weighted norm inequality involving convolutions of arbitrary analytic functions and certain confluent hypergeometric functions. This result implies a family of weighted norm inequalities both for entire functions of exponential type and for (generalized) hypergeometric series. The approach is based on author's general inequality for continuous functions and some hypergeometric transformations.

Leaf angle distribution is a key parameter to characterize canopy structure and plays a crucial role in controlling energy and mass balance in soil-vegetation-atmosphere-transfer system. Several leaf angle distribution functions found in literature have been proposed to account for the non-random distribution of leaf inclination angle with one or two parameters. In this paper, these leaf angle distribution functions (Beta distribution function, ellipsoidal function, rotated-ellipsoidal function, Verhoef's algorithm and de Wit's functions) were compared with field data collected in the First ISLSCP Field Experiment (FIFE) project and two sites within Ku-ring-gai Chase National Park, Sydney, Australia. All functions performed reasonably well. However, the comparison showed that the two-parameter functions including the Beta distribution function and Verhoef's algorithm commonly were more consistent predictors than one-parameter functions. G-statistics and chi(2) test applying to the estimates of leaf angle distribution demonstrated that Beta function presented more robustness over other functions, even the ellipsoidal leaf distribution function which has been widely used. Furthermore, the predictions of leaf angle distribution by these functions were used to calculate extinction coefficient and to separate foliage into sunlit and shaded parts. The results suggested that, ellipsoidal function may be suitable to be retrieved with remotely-sensed data and to compute extinction coefficient and fraction of sunlit foliage because this function requires only a single parameter, namely the ratio of the horizontal semi-axis length to the vertical semi-axis length of an ellipsoid. Finally, the comparison of three approaches (Nilson's, Fuchs' and Ross-Goudriaan's algorithms) for computing extinction coefficient indicated that, there was no significant difference between the three approaches. (c) 2006 Elsevier B.V. All rights reserved.

In this paper we study the mixed summation-integral type operators having Szasz and Beta basis functions. We extend the study of Gupta and Noor [V. Gupta, M.A. Noor, Convergence of derivatives for certain mixed Szasz–Beta operators, J. Math. Anal. Appl. 321 (1) (2006) 1–9] and obtain some direct results in local approximation without and with iterative combinations. In the last section are established direct global approximation theorems.

In the present paper, we study the rate of convergence in simultaneous approximation for the Bezier variant of the Baskakov-Beta operators by using the decomposition technique of functions of bounded variation.

We calculate the beta function of non-linear sigma models with S^{D+1} and AdS_{D+1} target spaces in a 1/D expansion up to order 1/D^2 and to all orders in \alpha'. This beta function encodes partial information about the spacetime effective action for the heterotic string to all orders in \alpha'. We argue that a zero of the beta function, corresponding to a worldsheet CFT with AdS_{D+1} target space, arises from competition between the one-loop and higher-loop terms, similarly to the bosonic and supersymmetric cases studied previously in hep-th/0512355. Various critical exponents of the non-linear sigma model are calculated, and checks of the calculation are presented., Comment: 36 pages, 7 figures

In this paper, the norm of an integral operator T : L r ( 0 , ∞ ) → L r ( 0 , ∞ ) ( r > 1 ) is obtained. As applications, a new bilinear integral operator inequality with the norm and the equivalent forms are given, and some new Hilbert's type inequalities with the best constant factors as the particular cases are established.

The sensitivity of the prediction of mean reaction rates in turbulent premixed flames to presumed PDF shape is studied. Three different presumed PDF shapes are considered: (i) a beta function PDF, (ii) a twin delta function PDF, and (iii) a PDF based on unstrained laminar flame properties. The unstrained laminar flame has the same thermochemistry as the turbulent flame. Emphasis is placed on capturing the finite rate chemistry effects and obtaining a simple expression for the mean reaction rate. It is shown that, as the PDFs approach their bimodal limit, the mean reaction rate expressions obtained using the above three PDFs reduce to a common form. These expressions differ only in the numerical value of a multiplying factor. Predictions are compared with DNS data. Under the conditions of this comparison, the beta function and twin delta function PDFs lead to significant errors, while the PDF based on properties of an unstrained laminar flame gives good agreement with the DNS.

The bosonic string theory evolved as an attempt to find a physical/quantum mechanical model capable of reproducing Euler’s beta function (Veneziano amplitude) and its multidimensional analogue. The multidimensional analogue of beta function was studied mathematically for some time from different angles by mathematicians such as Selberg, Weil and Deligne, among many others. The results of their studies apparently were not taken into account in physics literature on string theory. In a recent publication [IJMPA 19 (2004) 1655] an attempt was made to restore the missing links. The results of this publication are incomplete, however, since no attempts were made at reproduction of known spectra of both open an closed bosonic strings or at restoration of the underlying model(s) reproducing such spectra. Nevertheless, as discussed in this publication the existing mathematical interpretation of the multidimensional analogue of Euler’s beta function as one of the periods associated with the corresponding differential form “living” on the Fermat-type (hyper)surfaces, happens to be crucial for restoration of the quantum/statistical mechanical model reproducing such generalized beta function. Unlike the traditional formulations, this model is supersymmetric. Details leading to restoration of this model will be presented in the forthcoming Parts 2–4 of our work. They are devoted, respectively, to the group-theoretic, symplectic and combinatorial treatments of this model. In this paper the discussion is restricted mainly to the study of analytical properties of the multiparticle Veneziano and Veneziano-like (tachyon-free) amplitudes. In the last case, we demonstrate that the Veneziano-like amplitudes alone (with parameters adjusted accordingly) are capable of reproducing known spectra of both open and closed bosonic strings. The choice of parameters is subject to some constraints dictated by the mathematical interpretation of these amplitudes as periods of Fermat-type (hyper)surfaces considered as complex manifolds of Hodge-type.