Bayesian inference with information content model check for Langevin equations

The Bayesian data analysis framework has been proven to be a systematic and effective method of parameter inference and model selection for stochastic processes. In this work we introduce an information content model check which may serve as a goodness-of-fit, like the chi-square procedure, to complement conventional Bayesian analysis. We demonstrate this extended Bayesian framework on a system of Langevin equations, where coordinate dependent mobilities and measurement noise hinder the normal mean squared displacement approach.Comment: 10 pages, 7 figures, REVTeX, minor revision

Four-Fermion Limit of Gauge-Yukawa Theories

We elucidate and extend the conditions that map gauge-Yukawa theories at low energies into time-honoured gauged four-fermion interactions at high energies. These compositeness conditions permit to investigate theories of composite dynamics through gauge-Yukawa theories. Here we investigate whether perturbative gauge-Yukawa theories can have a strongly coupled limit at high-energy, that can be mapped into a four-fermion theory. Interestingly, we are able to precisely carve out a region of the perturbative parameter space supporting such a composite limit. This has interesting implications on our current view on models of particle physics. As a template model we use an $SU(N_C)$ gauge theory with $N_F$ Dirac fermions transforming according to the fundamental representation of the gauge group. The fermions further interact with a gauge singlet complex $N_F\times N_F$ Higgs that ceases to be a physical degree of freedom at the ultraviolet composite scale, where it gives away to the four-fermion interactions. We compute the hierarchy between the ultraviolet and infrared composite scales of the theory and show that they are naturally large and well separated. Our results show that some weakly coupled gauge-Yukawa theories can be viewed, in fact, as composite theories. It is therefore tantalising to speculate that the standard model, with its phenomenological perturbative Higgs sector, could hide, in plain sight, a composite theory.Comment: 20 pages, 9 figures, 10 pages Appendix, corrected typos and reference adde

Stable Extensions with(out) Gravity

We investigate the vacuum stability as well as the gravitational corrections in extensions of the Standard Model featuring a new complex scalar, and two Dirac fermions for different choices of the hypercharge of the scalar and one of the two fermions. The neutral fermion acquires loop-induced magnetic interactions with the Standard Model and could be identified with a dark matter candidate. To the lowest order in perturbation theory we show that these extensions can save the electroweak vacuum from being metastable. We then add the gravitational corrections to the different beta functions and discover that the models can be compatible with the asymptotically safe gravity scenario at the price of a heavier Higgs and lighter top mass.Comment: 12 pages, 4 figure

Trokuti s cjelobrojnim stranicama i trisektibilnim kutovima

Trokuti s cjelobrojnim stranicama i trisektibilnim kutovima posebna su klasa trokuta. Kosinusi kutova takvih trokuta su racionalni brojevi s dodatnim svojstvom da se mogu izraziti pomoću stanovitog polinoma 3. stupnja. Pokazuje se da ne postoje jednakostranični i jednakokračni pravokutni takvi trokuti. U ovom radu naglasak smo stavili na raznostranične trokute s cjelobrojnim stranicama i trisektibilnim kutovima, no osvrnuli smo se i na postojanje jednakokračnih koji nisu pravokutni te pravokutnih takvih trokuta. Pokazuje se da ključnu ulogu za postojanje razmatrane vrste trokuta ima tzv. rezidual, a to je pozitivni cijeli broj koji je na jednostavan način povezan s racionalnom vrijednosti kosinusa. Preciznije, dva šiljasta kuta s racionalnim kosinusima imaju isti rezidual ako i samo ako postoji trokut s cjelobrojnim stranicama koji sadrži ta dva kuta. Nadalje, pomoću reziduala dolazi se do opće formule za stranice promatranih raznostraničnih trokuta koji nemaju pravi kut. Glavni rezultat ovog rada otkriva da za svaki kvadratno slobodan pozitivan cijeli broj $r$ postoji beskonačno mnogo različitih trokuta s cjelobrojnim stranicama i trisektibilnim kutovima kojima je zajednički rezidual jednak $r$. U radu su izložene jednostavne metode za generiranje takvih trokuta te niz primjera za odgovarajuće duljine triju stranica. Konačno, pokazuje se da su šiljasti kutovi primitivnog Pitagorinog trokuta su trisektibilni ako i samo ako je hipotenuza potpuni kub.Integer-sided triangles with trisectible angles are a special class of triangles. Cosines of angles of such triangles are rational numbers with the additional property of representation by a specific third degree polynomial. It is shown that no equilateral triangles and no right angled isosceles triangles belong to that class. In this paper, the emphasis is placed on scalene integer-sided triangles with trisectible angles, but the existence of right triangles with these properties is thoroughly investigated, too, as well as the inclusion of isosceles triangles without a right angle in that class. It is shown that the key feature for the existence of the considered triangle type is the so-called residual. Residual is a positive integer that is in a simple way associated to the rational value of the cosine. More precisely, two acute angles with rational cosines have the same residual if and only if there is an integer-sided triangle containing these two angles. Furthermore, a general formula based on residuals is derived for the sidelengths of the observed scalene triangles without a right angle. The main result of this paper reveals that for each square-free positive integer r there exist infinitely many distinct scalene integersided triangles with trisectible angles and with r as their common residual. Some simple methods for generating such triangles are presented and examples for the corresponding sidelengths are given. Finally, it is shown that the acute angles of primitive Pythagorean triangles are trisectible if and only if their hypotenuse is a perfect cube

pryslied vir ’n vreeslose gewete

A poem in Afrikaans, English, isiXhosa and Sesotho in honour of Desmond Mpilo Tutu (7 October 1931 – 26 December 2021)