In this paper we consider modification of general relativity extending R−2Λ by nonlocal term of the formis an analytic function of the d'Alembert operator ✷. We have found some exact cosmological solutions of the corresponding equations of motion without matter and with Λ = 0. One of these solutions is a(t) = A t 2 3 e Λ 14 t 2 , which imitates properties similar to an interplay of the dark matter and the dark energy. For this solution we calculated some cosmological parameters which are in a good agreement with observations. This nonlocal gravity model has not the Minkowski space solution. We also found several conditions which function F (✷) has to satisfy.

A new flavonoid glucoside derivative, patuletin 3-O-(2-O-feruloyl)-β-D-glucuronopyranosyl-(1→2)-β-Dglucopyranoside, named atriplexin IV (1), and three new triterpenoid saponin derivatives, two sulfonylated, β-Dglucopyranosyl-3-O-(2-O-sulfo-β-D-galactopyranosyl)-(1→2)-α-L-arabinopyranoside-30-alolean-12-en-28-oate (2), named atriplexogenin I, β-D-glucopyranosyl-3-O-(2-O-sulfo-β-D-galactopyranosyl)-(1→2)-α-L-arabinopyranoside)-30-hydroxyolean-12-en-28-oate (3), named atriplexogenin II, and β-D-glucopyranosyl-3-O-(β-D-glucopyranosyl-(1→2)-β-D-galactopyranosyl-(1→2)-α-L-arabinopyranoside)-30-alolean-12-en-28-oate (4), named atriplexogenin III, were isolated by silica gel column and semipreparative HPLC chromatography from the n-butanol extract of the salt marsh plant Atriplex tatarica. In addition, two known secondary metabolites, patuletin3-O-β-D-apiofuranosyl-(1‴→2″)-β-D-glucopyranoside (5) and patuletin 3-O-5‴-Oferuloyl-β-D-apiofuranosyl-(1‴→2″)-β-D-glucopyranoside ( 6), were isolated for the first time from A. tatarica. The structures of the isolated compounds were elucidated by 1D and 2D NMR, HRESIMS, IR, and UV data. Antibacterial activity by the microdilution method and antibiofilm activity against P. aeruginosa were assessed. Compound 5 possesses significant antibacterial activity, while the most potent antibiofilm agent is compound 2.

We consider nonlocal modified Einstein gravity without matter, where nonlocal term has the form P (R)F ( )Q(R). For this model, in this paper we give the derivation of the equations of motion in detail. This is not an easy task and presented derivation should be useful to a researcher who wants to investigate nonlocal gravity. Also, we present the second variation of the related Einstein-Hilbert modified action and basics of gravity perturbations.

In this paper, we investigate a nonlocal modification of general relativity (GR) with action S = 1 16 π G ∫ [ R − 2 Λ + ( R − 4 Λ ) F ( □ ) ( R − 4 Λ ) ] − g d 4 x , where F ( □ ) = ∑ n = 1 + ∞ f n □ n is an analytic function of the d’Alembertian □. We found a few exact cosmological solutions of the corresponding equations of motion. There are two solutions which are valid only if Λ ≠ 0 , k = 0 , and they have no analogs in Einstein’s gravity with cosmological constant Λ . One of these two solutions is a ( t ) = A t e Λ 4 t 2 , that mimics properties similar to an interference between the radiation and the dark energy. Another solution is a nonsingular bounce one a ( t ) = A e Λ t 2 . For these two solutions, some cosmological aspects are discussed. We also found explicit form of the nonlocal operator F ( □ ) , which satisfies obtained necessary conditions.

We consider a modification of GR with a special type of a non-local f (R). The structure of the non-local operators is motivated by the string field theory and p-adic string theory. The spectrum is derived explicitly and the ghost-free condition for the model is formulated. We pay special attention to the classical stability of the de Sitter solution in our model and formulate the conditions on the model parameters to have a stable configuration. Relevance of unstable configurations for the description of the de Sitter phase during inflation is specifically discussed.