Perturbative renormalization of the $$ \mathrm{T}\overline{\mathrm{T}} $$-deformed free massive Dirac fermion

In this paper we explicitly carry out the perturbative renormalization of the $$ T\overline{T} $$ T T ¯ -deformed free massive Dirac fermion in two dimensions up to second order in the coupling constant. This is done by computing the two-to-two S-matrix using the LSZ reduction formula and canceling out the divergences by introducing counterterms. We demonstrate that the renormalized Lagrangian is unambiguously determined by demanding that it gives the correct S-matrix of a $$ T\overline{T} $$ T T ¯ -deformed integrable field theory. Remarkably, the renormalized Lagrangian is qualitatively very different from its classical counterpart.

A STUDY OF REDUCTION FORMULA FOR GENERALISED H-FUNCTION OF TWO VARIABLES

In the present paper, the author has introduced reduction formula for Generalised H-Function of two variables. This paper deals with certain identities and reduction formulae for generalised H-Function, which are of great interest and generalise many known and unknown results in literature, especially the results given by Srivastava [1] and Cook [2]. Also some interesting special cases of H- Functions have been taken in which H-Function of two variables has reduced to one variable. Key Words: H-Function, Hyper geometric Function, Mellin Transform, H-Function of one and two variables.

A NEW PROOF OF A REDUCTION FORMULA FOR THE APPELL SERIES F3 DUE TO BAILEY

In this short note, we provide a new proof of an interesting and useful reduction formula for the Appell series $F_{3}$ due to Bailey [{\it On the sum of a terminating ${}_3F_2(1)$}, Quart. J. Math. Oxford Ser. (2) {\bf4} (1953), 237--240].

Calculation Approaches of the Probable Maximum Discharge of Spring Flood at Ungauged Sites in the Southern Buh River Basin, Ukraine

Calculation of probable maximum discharge of spring flood are the great practical importance, since it is the basis to plan and design of different hydraulic structures, such as dams, culverts, urban and agriculture drainage systems, etc. Thus, the updating of the methodical approaches and parameters of the empirical formulas which using in the determining of the probable maximum discharge of spring flood at ungauged sites of the river basin is an actual task. In this paper for the Southern Buh River Basin were updated the parameters of the reduction formula and the limiting intensity formula of streamflow which are using to calculated of the probable maximum discharge of spring flood at ungauged basin in Ukraine. The presented results illustrate that parameters of empirical formulas that were calculated according to modern observation series (since the beginning of the observations to 2010) in comparison with previously received (since the beginning of the observations to 1980) have significant changes. We found out that it is due to cyclical of the long-term fluctuations of the maximum streamflow of spring flood in the Southern Buh River Basin. We also illustrated that for the small ungauged basins have the difficulties with the choice of rivers-analogues.

Bounds related to projective equivalence classes of good filtrations

Let [Formula: see text] be a regular ideal in noetherian ring [Formula: see text]. Mc Adam and Ratliff showed the existence of the unique minimal reduction number of [Formula: see text], noted [Formula: see text], such that for every minimal reduction [Formula: see text] of [Formula: see text], [Formula: see text] and [Formula: see text]. They showed that the set of integers [Formula: see text] is bounded in terms of the analytic spread of [Formula: see text]. Here, we extend these results to good filtrations. Let [Formula: see text] be a good filtration on [Formula: see text], we show that the set of integers [Formula: see text] is bounded.

Scattering processes

The unitarity of the S-matrix is used to derive the optical theorem. The connection between Green functions and scattering amplitudes given by the LSZ reduction formula is derived. The trace and the helicity method are developed and applied to the calculation of QED processes. The emission of soft photons and gravitons is discussed. In an appendix, the connection between S-matrix elements, Feynman amplitudes and decay rates or cross-sections is derived.