Diagonal Map Research Articles

On two notions of a gerbe over a stack

Let G be a Lie groupoid. The category BG of principal G-bundles defines a differentiable stack. On the other hand, given a differentiable stack D, there exists a Lie groupoid H such that BH is isomorphic to D. Define a gerbe over a stack as a morphism of stacks F:D→C, such that F and the diagonal map ΔF:D→D×CD are epimorphisms. This paper explores the relationship between a gerbe defined above and a Morita equivalence class of a Lie groupoid extension.

Vehicle Re-Identification Using Quadruple Directional Deep Learning Features

In order to resist the adverse effect of viewpoint variations for improving vehicle re-identification performance, we design quadruple directional deep learning networks to extract quadruple directional deep learning features (QD-DLF) of vehicle images. The quadruple directional deep learning networks are with similar overall architecture, including the same basic deep learning architecture but different directional feature pooling layers. Specifically, the same basic deep learning architecture is a shortly and densely connected convolutional neural network to extract basic feature maps of an input square vehicle image in the first stage. Then, the quadruple directional deep learning networks utilize different directional pooling layers, i.e., horizontal average pooling (HAP) layer, vertical average pooling (VAP) layer, diagonal average pooling (DAP) layer and anti-diagonal average pooling (AAP) layer, to compress the basic feature maps into horizontal, vertical, diagonal and anti-diagonal directional feature maps, respectively. Finally, these directional feature maps are spatially normalized and concatenated together as a quadruple directional deep learning feature for vehicle re-identification. Extensive experiments on both VeRi and VehicleID databases show that the proposed QD-DLF approach outperforms multiple state-of-the-art vehicle re-identification methods.

The transfer is functorial

We prove that the Becker–Gottlieb transfer is functorial up to homotopy, for all fibrations with finitely dominated fibers. This resolves a lingering foundational question about the transfer, which was originally defined in the late 1970s in order to simplify the proof of the Adams conjecture. Our approach differs from previous attempts in that we closely emulate the geometric argument in the case of a smooth fiber bundle. This leads to a “multiplicative” description of the transfer, different from the standard presentation as the trace of a diagonal map.

On a Sufficient Condition for Weak Sharp Efficiency in Multiobjective Optimization

In this paper, we provide sufficient conditions entailing the existence of weak sharp efficient points of a multiobjective optimization problem. The approach uses variational analysis techniques, like regularity and subregularity of the diagonal subdifferential map related to a suitable scalar equilibrium problem naturally associated to the multiobjective optimization problem.

On the fundamental class of an essentially smooth scheme-map

Let f: X -> Z be a separated essentially-finite-type flat map of noetherian schemes, and \delta: X --> X \times_Z X the diagonal map. The fundamental class C_f (globalizing residues) is a map from the relative Hochschild functor L\delta^*\delta_* f^* to the relative dualizing functor f^! A compatibility between this C_f and derived tensor product is shown. The main result is that, in a suitable sense, C_f generalizes Verdier's classical isomorphism for smooth f with fibers of dimension d, an isomorphism that binds f^! to relative d-forms.

On the Hochschild cohomology ring of integral cyclic algebras

We determine the ring structure of the Hochschild cohomology HH*(Γ) of an integral cyclic algebra Γ by giving a projective bimodule resolution of Γ and calculating cup product by means of a diagonal approximation map.

Loop homological invariants associated to real projective spaces

Let A be a based subspace of Y. Under the assumptions that Y is path-connected and that the reduced diagonal map of A induces the zero map in all mod 2 reduced homology groups, we compute a formula for the mod 2 reduced Poincaré series of the loop space Ω((A∧RP∞)∪A∧RP1(Y∧RP1)). Here RP∞ and RP1 denote the infinite real projective space and the real projective line respectively.

The topological complexity and the homotopy cofiber of the diagonal map for non-orientable surfaces

We show that the Lusternik-Schnirelmann category of the homotopy cofiber of the diagonal map of non-orientable surfaces equals three. Also, we prove that the topological complexity of non-orientable surfaces of genus > 4 > 4 is four.

The ghost length and duality on the chain and cochain type levels

We establish equalities between cochain and chain type levels of maps by making use of exact functors which connect appropriate derived and coderived categories. Relevant conditions for levels of maps to be finite are extracted from the equalities which we call duality on the levels. Moreover, we give a lower bound of the cochain type level of the diagonal map on the classifying space of a Lie group by considering the ghostness of a shriek map which appears in derived string topology. A variant of Koszul duality for a differential graded algebra is also discussed.

Bott–Kitaev periodic table and the diagonal map

Building on the ten-way symmetry classification of disordered fermions, the authors have recently given a homotopy-theoretic proof of Kitaev's ‘periodic table’ for topological insulators and superconductors. The present paper offers an introduction to the physical setting and the mathematical model used. Basic to the proof is the so-called diagonal map, a natural transformation akin to the Bott map of algebraic topology, which increases by one unit both the momentum-space dimension and the symmetry index of translation-invariant ground states of gapped free-fermion systems. This mapping is illustrated here with a few examples of interest. (Based on a talk delivered by the senior author at the Nobel Symposium on ‘New Forms of Matter: Topological Insulators and Superconductors’; Stockholm, 13–15 June, 2014.)

The Lifting-Extension Problem for Duchain Complexes of Dwyer and Kan

Abstract A duchain complex of W. Dwyer and D. Kan is a common extension of the notions of a chain complex and a cochain complex. Given a square commutative diagram of duchain complexes, the lifting-extension problem asks whether there exists a diagonal map making the two resulting triangles commute. Duchain complexes have a model category structure, and hence a lift exists if the left vertical map is a cofibration, the right vertical map is a fibration, and one of them is a weak equivalence. We show that it is possible to replace the two conditions above, by a countably infinite, bigraded, family of conditions which guarantee the existence of a lift.

Robust differential modulations for asynchronous cooperative systems

Accurate channel state information and time/frequency synchronization are challenging to acquire in mobile ad hoc cooperative set-ups, where resource-constrained relaying terminals rapidly (dis)join cooperation. This paper derives low-complexity differential modulations and cooperative transmission schemes so that detection at the receiver is feasible without channel knowledge or synchronization information. A simple distributed doubly-differential (DD) time-division multiplexing (TDM) scheme using diagonal space–time (ST) unitary codes is derived to bypass multiple carrier frequency offsets (CFOs) and collect space diversity. Precoded differential orthogonal frequency division multiplexing (OFDM) transmissions are derived to additionally bypass timing offsets and collect multipath diversity using either DD diagonal or single-differential (SD) orthogonal ST block codes (OSTBCs). DD diagonal mappings suffer from considerable coding gain and signal-to-noise ratio (SNR) loss due to unitary constellation constraints and noise enhancement at the receiver. SD-OSTBCs achieve higher performance with reduced decoding complexity, at the cost of reduced CFO mitigation range. Simple and robust non-coherent selective and adaptive transmissions are also included, showing that full space and multipath diversity can still be achieved even with decoding errors at relays. Simulations corroborate theoretical claims.

Relative category and monoidal topological complexity

If a map f has a homotopy retraction, then Doeraene and El Haouari conjectured that the sectional category and the relative category of f are the same. In this work we discuss this conjecture for some lower bounds of these invariants. In particular, when we consider the diagonal map, we obtain some results supporting Iwase–Sakai's conjecture which asserts that the topological complexity is the monoidal topological complexity.

Estimation of Optimized Energy and Latency Constraint for Task Allocation in 3d Network on Chip

In Network on Chip (NoC) rooted system, energy consumption is affected by task scheduling and allocation schemes which affect the performance of the system. In this paper we test the pre-existing proposed algorithms and introduced a new energy skilled algorithm for 3D NoC architecture. An efficient dynamic and cluster approaches are proposed along with the optimization using bio-inspired algorithm. The proposed algorithm has been implemented and evaluated on randomly generated benchmark and real life application such as MMS, Telecom and VOPD. The algorithm has also been tested with the E3S benchmark and has been compared with the existing mapping algorithm spiral and crinkle and has shown better reduction in the communication energy consumption and shows improvement in the performance of the system. On performing experimental analysis of proposed algorithm results shows that average reduction in energy consumption is 49%, reduction in communication cost is 48% and average latency is 34%. Cluster based approach is mapped onto NoC using Dynamic Diagonal Mapping (DDMap), Crinkle and Spiral algorithms and found DDmap provides improved result. On analysis and comparison of mapping of cluster using DDmap approach the average energy reduction is 14% and 9% with crinkle and spiral.

Mean dimension and AH-algebras with diagonal maps

Mean dimension for AH-algebras with diagonal maps is introduced. It is shown that if a simple unital AH-algebra with diagonal maps has mean dimension zero, then it has strict comparison on positive elements. In particular, the strict order on projections is determined by traces. Moreover, a lower bound of the mean dimension is given in term of Toms' comparison radius. Using classification results, if a simple unital AH-algebra with diagonal maps has mean dimension zero, it must be an AH-algebra without dimension growth. Two classes of AH-algebras with diagonal maps are shown to have mean dimension zero: the class of AH-algebras with at most countably many extremal traces, and the class of AH-algebras with numbers of extreme traces which induce same state on the K0-group being uniformly bounded (in particular, this class includes AH-algebras with real rank zero).

Diagonal Mapping in Anisotropic Spaces of Analytic Functions in Polydisc with Mixed Norm

We give a complete characterization of the analytic functions in the unit disk, which are the restriction to the diagonal in the polydisc of the holomorphic functions from the anisotropic weighted spaces with mixed norm.

Homology decompositions of the loops on 1–stunted Borel constructions ofC2–actions

Carlsson's construction is a simplicial group whose geometric realization is the loop space of the 1-stunted reduced Borel construction. Our main results are: i) Given a pointed simplicial set acted upon by the discrete cyclic group C_2 of order 2, if the orbit projection has a section, then this loop space has a mod 2 homology decomposition; ii) If the reduced diagonal map of the C_2-invariant set is homologous to zero, then the pinched sets in the above homology decomposition themselves have homology decompositions in terms of the C_2-invariant set and the orbit space. Result i) generalizes a previous homology decomposition of the second author for trivial actions. To illustrate these two results, we completely compute the mod 2 Betti numbers for an example.

Entanglement and output entropy of the diagonal map

We review some properties of the convex roof extension, a construction used, e.g., in the definition of the entanglement of formation. Especially we consider the use of symmetries of channels and states for the construction of the convex roof. As an application we study the entanglement entropy of the diagonal map for permutation symmetric real $N=3$ states $\omega(z)$ and solve the case $z<0$ where $z$ is the non-diagonal entry in the density matrix. We also report a surprising result about the behavior of the output entropy of the diagonal map for arbitrary dimensions $N$; showing a bifurcation at $N=6$.

Up-to-one approximations of sectional category and topological complexity

Jamesʼ sectional category and Farberʼs topological complexity are studied in a general and unified framework.We introduce ‘relative’ and ‘strong relative’ forms of the category for a map and show that both can differ from sectional category by just one. A map has sectional or relative category at most n if, and only if, it is ‘dominated’ in a (different) sense by a map with strong relative category at most n. A homotopy pushout can increase sectional category but neither homotopy pushouts, nor homotopy pullbacks, can increase (strong) relative category. This makes (strong) relative category a convenient tool to study sectional category. We completely determine the sectional and relative categories of the fibres of the Ganea fibrations.In particular, the ‘topological complexity’ of a space is the sectional category of the diagonal map, and so it can differ from the (strong) relative category of the diagonal by just one. We call the strong relative category of the diagonal ‘strong complexity’. We show that the strong complexity of a suspension is at most two.

Mining Dense Patterns from Off Diagonal Protein Contact Maps

The three dimensional structure of proteins is useful to carry out the biophysical and biochemical functions in a cell. Protein contact maps are 2D representations of contacts among the amino acid residues in the folded protein structure. Proteins are biochemical compounds consisting of one or more polypeptides, facilitating a biological function. Many researchers make note of the way secondary structures are clearly visible in the contact maps where helices are seen as thick bands and the sheets as orthogonal to the diagonal. In this paper, we explore several machine learning algorithms to data driven construction of classifiers for assigning protein off diagonal contact maps. A simple and computationally inexpensive algorithm based on triangle subdivision method is implemented to extract twenty features from off diagonal contact maps. This method successfully characterizes the offdiagonal interactions in the contact map for predicting specific folds. NaiveBayes, J48 and REPTree classification results with Recall 76.38%, 91.66% and 80.32% are obtained respectively. General Terms Protein Contact Map.