342 research outputs found
Off-diagonal cosmological solutions in emergent gravity theories and Grigory Perelman entropy for geometric flows
We develop an approach to the theory of relativistic geometric flows and
emergent gravity defined by entropy functionals and related statistical
thermodynamics models. Nonholonomic deformations of G. Perelman's functionals
and related entropic values are used for deriving relativistic geometric
evolution flow equations. For self-similar configurations, such equations
describe generalized Ricci solitons defining modified Einstein equations. We
analyze possible connections between relativistic models of nonholonomic Ricci
flows and emergent modified gravity theories. We prove that corresponding
systems of nonlinear partial differential equations, PDEs, for entropic flows
and modified gravity possess certain general decoupling and integration
properties. There are constructed new classes of exact and parametric solutions
for nonstationary configurations and locally anisotropic cosmological metrics
in modified gravity theories and general relativity. Such solutions describe
scenarios of nonlinear geometric evolution and gravitational and matter field
dynamics with pattern-forming and quasiperiodic structure and various space
quasicrystal and deformed spacetime crystal models. We analyze new classes of
generic off-diagonal solutions for entropic gravity theories and show how such
solutions can be used for explaining structure formation in modern cosmology.
Finally, we speculate why the approaches with Perelman-Lyapunov type
functionals are more general or complementary to the constructions elaborated
using the concept of Bekenstein-Hawking entropy.Comment: accepted to EPJC; latex2e 11pt, 35 pages with a table of contents; v3
is substantially modified with a new title and a new co-autho
Nonassociative Ricci flows, star product and R-flux deformed black holes, and swampland conjectures
We extend to a theory of nonassociative geometric flows a string-inspired
model of nonassociative gravity determined by star product and R-flux
deformations. The nonassociative Ricci tensor and curvature scalar defined by
(non) symmetric metric structures and generalized (non) linear connections are
used for defining nonassociative versions of Grigori Perelman F- and
W-functionals for Ricci flows and computing associated thermodynamic variables.
We develop and apply the anholonomic frame and connection deformation method,
AFCDM, which allows us to construct exact and parametric solutions describing
nonassociative geometric flow evolution scenarios and modified Ricci soliton
configurations with quasi-stationary generic off-diagonal metrics. There are
provided explicit examples of solutions modelling geometric and statistical
thermodynamic evolution on a temperature-like parameter of modified black hole
configurations encoding nonassociative star-product and R-flux deformation
data. Further perspectives of the paper are motivated by nonassociative
off-diagonal geometric flow extensions of the swampland program, related
conjectures and claims on geometric and physical properties of new classes of
quasi-stationary Ricci flow and black hole solutions.Comment: 81 pages latex2e 11pt, v1 accepted to Fort. der Physik, FP; 5th
partner work to arXiv: 2106.01320, 2106.01869, 2108.04689,2207.05157
published in FP, FP, EPJC, JHE
Hardness of Covering Alignment : Phase Transition in Post-Sequence Genomics
Covering alignment problems arise from recent developments in genomics; so called pan-genome graphs are replacing reference genomes, and advances in haplotyping enable full content of diploid genomes to be used as basis of sequence analysis. In this paper, we show that the computational complexity will change for natural extensions of alignments to pan-genome representations and to diploid genomes. More broadly, our approach can also be seen as a minimal extension of sequence alignment to labelled directed acyclic graphs (labeled DAGs). Namely, we show that finding a covering alignment of two labeled DAGs is NP-hard even on binary alphabets. A covering alignment asks for two paths R-1 (red) and G(1) (green) in DAG D-1 and two paths R-2 (red) and G(2) (green) in DAG D-2 that cover the nodes of the graphs and maximize the sum of the global alignment scores: asosp(R-1), sp(R-2)) + asosp(G(1)), sp(G(2))), where sp(P) is the concatenation of labels on the path P. Pair-wise alignment of haplotype sequences forming a diploid chromosome can be converted to a two-path coverable labelled DAG, and then the covering alignment models the similarity of two diploids over arbitrary recombinations. We also give a reduction to the other direction, to show that such a recombination-oblivious diploid alignment is NP-hard on alphabets of size 3.Peer reviewe
Sparse Dynamic Programming on DAGs with Small Width
The minimum path cover problem asks us to find a minimum-cardinality set of paths that cover all the nodes of a directed acyclic graph (DAG). We study the case when the size k of a minimum path cover is small, that is, when the DAG has a small width. This case is motivated by applications in pan-genomics, where the genomic variation of a population is expressed as a DAG. We observe that classical alignment algorithms exploiting sparse dynamic programming can be extended to the sequence-against-DAG case by mimicking the algorithm for sequences on each path of a minimum path cover and handling an evaluation order anomaly with reachability queries. Namely, we introduce a general framework for DAG-extensions of sparse dynamic programming. This framework produces algorithms that are slower than their counterparts on sequences only by a factor k. We illustrate this on two classical problems extended to DAGs: longest increasing subsequence and longest common subsequence. For the former, we obtain an algorithm with running time O(k vertical bar E vertical bar log vertical bar V vertical bar). This matches the optimal solution to the classical problem variant when the input sequence is modeled as a path. We obtain an analogous result for the longest common subsequence problem. We then apply this technique to the co-linear chaining problem, which is a generalization of the above two problems. The algorithm for this problem turns out to be more involved, needing further ingredients, such as an FM-index tailored for large alphabets and a two-dimensional range search tree modified to support range maximum queries. We also study a general sequence-to-DAG alignment formulation that allows affine gap costs in the sequence. The main ingredient of the proposed framework is a new algorithm for finding a minimum path cover of a DAG (V, E) in O(k vertical bar E vertical bar log vertical bar V vertical bar) time, improving all known time-bounds when k is small and the DAG is not too dense. In addition to boosting the sparse dynamic programming framework, an immediate consequence of this new minimum path cover algorithm is an improved space/time tradeoff for reachability queries in arbitrary directed graphs.Peer reviewe
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