Heat Kernels on the AdS(2) cone and Logarithmic Corrections to Extremal Black Hole Entropy

We develop new techniques to efficiently evaluate heat kernel coefficients for the Laplacian in the short-time expansion on spheres and hyperboloids with conical singularities. We then apply these techniques to explicitly compute the logarithmic contribution to black hole entropy from an N=4 vector multiplet about a Z(N) orbifold of the near-horizon geometry of quarter--BPS black holes in N=4 supergravity. We find that this vanishes, matching perfectly with the prediction from the microstate counting. We also discuss possible generalisations of our heat kernel results to higher-spin fields over Z(N) orbifolds of higher-dimensional spheres and hyperboloids.Comment: 41 page

Logarithmic Corrections to Extremal Black Hole Entropy in N = 2, 4 and 8 Supergravity

We compute the logarithmic correction to black hole entropy about exponentially suppressed saddle points of the Quantum Entropy Function corresponding to Z(N) orbifolds of the near horizon geometry of the extremal black hole under study. By carefully accounting for zero mode contributions we show that the logarithmic contributions for quarter--BPS black holes in N=4 supergravity and one--eighth BPS black holes in N=8 supergravity perfectly match with the prediction from the microstate counting. We also find that the logarithmic contribution for half--BPS black holes in N = 2 supergravity depends non-trivially on the Z(N) orbifold. Our analysis draws heavily on the results we had previously obtained for heat kernel coefficients on Z(N) orbifolds of spheres and hyperboloids in arXiv:1311.6286 and we also propose a generalization of the Plancherel formula to Z(N) orbifolds of hyperboloids to an expression involving the Harish-Chandra character of SL(2,R), a result which is of possible mathematical interest.Comment: 40 page

Maximal area integral problem for certain class of univalent analytic functions

One of the classical problems concerns the class of analytic functions $f$ on the open unit disk $|z|<1$ which have finite Dirichlet integral $\Delta(1,f)$, where $$\Delta(r,f)=\iint_{|z|<r}|f'(z)|^2 \, dxdy \quad (0<r\leq 1).$$ The class ${\mathcal S}^*(A,B)$ of normalized functions $f$ analytic in $|z|<1$ and satisfies the subordination condition $zf'(z)/f(z)\prec (1+Az)/(1+Bz)$ in $|z|<1$ and for some $-1\leq B\leq 0$, $A\in {\mathbb C}$ with $A\neq B$, has been studied extensively. In this paper, we solve the extremal problem of determining the value of $$\max_{f\in {\mathcal S}^*(A,B)}\Delta(r,z/f)$$ as a function of $r$. This settles the question raised by Ponnusamy and Wirths in [11]. One of the particular cases includes solution to a conjecture of Yamashita which was settled recently by Obradovi\'{c} et. al [9].Comment: 16 pages, 8 figures, 3 table

Opterećenje tekućinom i prijenosna sposobnost bubrega u dvogrbe deve u dehidraciji i rehidraciji zimi i ljeti

The effect of dehydration and rehydration was studied during winter and summer on solute loads and transfer function of kidney in healthy adult female dromedary camels. Kidney solute loads (KSLs) which included plasma loads (PL) and tubular loads (TL) were determined for glucose, proteins, urea, creatinine, sodium, potassium, chloride, calcium and phosphorus. The dehydration period was of 24 days in winter and 13 days in summer. Water was provided ad libitum during control and rehydration periods and was restricted completely during dehydration period. The mean value of TFK during summer control was significantly (P≤0.05) lower than that in winter control. In winter the mean values of TFK during rehydration phases differed significantly (P≤0.05) from control values. A similar trend was observed during summer, except that the calculations for TFK could not be made at hour ½ and at hour 2 of rehydration since animals did not void urine. During dehydration periods in both seasons PL and TL mean values decreased significantly (P≤0.05) from respective control mean values. It was concluded that during dehydration reduction in kidney solute loads was indicative of the water conservation ability of camels because reduced plasma loads and tubular loads resulted in trapping of constituents in the plasma to hold more water.Istražen je učinak dehidracije i rehidracije u tijeku zime i ljeta na opterećenje tekućinom i prijenosnu sposobnost bubrega u zdravih ženki dvogrbe deve. Opterećenje obuhvaća plazmalno (PO) i tubularno (TO) opterećenje, a određivano je za glukozu, bjelančevine, mokraćevinu, kreatinin, natrij, kalij, klor, kalcij i fosfor. Razdoblje dehidracije trajalo je 24 dana zimi i 13 dana ljeti. Životinje su po volji pile vodu tijekom kontrolnoga i rehidracijskoga razdoblja, ali vodu nisu dobivale u tijeku dehidracijskoga razdoblja. Srednja vrijednost prijenosne sposobnosti bubrega u tijeku ljetnih mjeseci bila je značajno niža (P≤0,05) u odnosu na zimsko kontrolno razdoblje. Zimi su se srednje vrijednosti prijenosne sposobnosti bubrega za vrijeme rehidracije značajno razlikovale u odnosu na kontrolne vrijednosti (P≤0,05). Sličan je trend zabilježen ljeti, osim što izračuni nisu mogli biti učinjeni sat i pol te dva sata nakon rehidracije, jer životinje nisu izlučivale mokraću. Srednje vrijednosti PO i TO bile su značajno manje u odnosu na kontrolu (P≤0,05) u tijeku dehidracijskoga razdoblja u oba godišnja doba. Zaključuje se da je smanjeno opterećenje bubrega u tijeku dehidracije dobar pokazatelj sposobnosti čuvanja vode s obzirom na to da smanjeno plazmalno i tubularno opterećenje dovodi do zadržavanja sastojaka u plazmi koji imaju sposobnost osmotskoga zadržavanja vode