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In this paper we develop a novel mathematical formalism for the modeling ofneural information networks endowed with additional structure in the form ofassignments of resources, either computational or metabolic or informational.The starting point for this construction is the notion of summing functors andof Segal's Gamma-spaces in homotopy theory. The main results in this paperinclude functorial assignments of concurrent/distributed computingarchitectures and associated binary codes to networks and their subsystems, acategorical form of the Hopfield network dynamics, which recovers the usualHopfield equations when applied to a suitable category of weighted codes, afunctorial assignment to networks of corresponding information structures andinformation cohomology, and a cohomological version of integrated information. 

We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with valence-4internal vertices, relevant to CTKT tensor models. The results hint at a complex and interesting structure in terms of divisibility relations or nontrivial relations between classes of graphs in different families. Using the recursive relations, we prove that the Grothendieck classes of all melonic graphs are positive as polynomials in the class of the moduli space\mathcal M_{0,4}. We also conjecture that the corresponding polynomials arelog-concave, on the basis of hundreds of explicit computations. 

We analyze the interplay between contact geometry and Gabor filters signalanalysis in geometric models of the primary visual cortex. We show inparticular that a specific framed lattice and an associated Gabor system isdetermined by the Legendrian circle bundle structure of the $$3$$-manifold ofcontact elements on a surface (which models the V1-cortex), together with thepresence of an almost-complex structure on the tangent bundle of the surface(which models the retinal surface). We identify a scaling of the lattice, alsodictated by the manifold geometry, that ensures the frame condition issatisfied. We then consider a $$5$$-dimensional model where receptor profilesalso involve a dependence on frequency and scale variables, in addition to thedependence on position and orientation. In this case we show that a proposedprofile window function does not give rise to frames (even in a distributionalsense), while a natural modification of the same generates Gabor frames withrespect to the appropriate lattice determined by the contact geometry. 

A bstract We introduce a unifying framework for the construction of holographic tensor networks, based on the theory of hyperbolic buildings. The underlying dualities relate a bulk space to a boundary which can be homeomorphic to a sphere, but also to more general spaces like a Menger sponge type fractal. In this general setting, we give a precise construction of a large family of bulk regions that satisfy complementary recovery. For these regions, our networks obey a Ryu-Takayanagi formula. The areas of Ryu-Takayanagi surfaces are controlled by the Hausdorff dimension of the boundary, and consistently generalize the behavior of holographic entanglement entropy in integer dimensions to the non-integer case. Our construction recovers HaPPY-like codes in all dimensions, and generalizes the geometry of Bruhat-Tits trees. It also provides examples of infinite-dimensional nets of holographic conditional expectations, and opens a path towards the study of conformal field theory and holography on fractal spaces. 

Abstract We consider cosmological models based on the spectral action formulation of (modified) gravity. We analyze the coupled effects, in this model, of the presence of nontrivial cosmic topology and of fractality in the large scale structure of spacetime. We show that the topology constrains the possible fractal structures, and in turn the correction terms to the spectral action due to fractality distinguish the various cosmic topology candidates, with effects detectable in a slow-roll inflation scenario, through the power spectra of the scalar and tensor fluctuations. We also discuss explicit effects of the presence of fractal structures on the gravitational waves equations.