Spaces ofp-vectors of bounded rank

Gelbord, Boaz; Meshulam, Roy

doi:10.1007/bf02784151

Cited by 5 publications

(3 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Rather than deciding by fiat that the CB model is wrong, we would conclude that it also offers an explanation for the extremely narrow jets observed at X-ray frequencies, not only in Pictor A, but in more than twenty other quasars, blazars and radio galaxies that were observed with Chandra (Gelbord et al 2003, Marshall et al 2003. In several cases the narrow jets were resolved into plasmoids (CBs), such as in M87, Cygnus A and 3C279, or in microquasars such as GRS 1915+105 (Dhawan Mirabel & Rodriguez 2000), XTE J1550-564 (Corbel et al 2002), 4U 1755-33 (Angelini & White, 2003, and even in pulsars (e.g.…”

Section: Appendix Ii: Ags In the Fb Models And In The Cb Modelmentioning

confidence: 91%

Towards a complete theory of gamma-ray bursts

2004

View full text Add to dashboard Cite

Gamma Ray Bursts (GRBs) are notorious for their diversity. Yet, they have a series of common features. The typical energy of their γ rays is a fraction of an MeV. The energy distributions are well described by a "Band spectrum", with "peak energies" spanning a surprisingly narrow range. The time structure of a GRB consists of pulses, superimposed or not, rising and decreasing fast. The number of photons in a pulse, the pulses' widths and their total energy vary within broad but specific ranges. Within a pulse, the energy spectrum softens with increasing time. The duration of a pulse decreases at higher energies and its peak intensity shifts to earlier time. Many other correlations between pairs of GRB observables have been identified. Last (and based on one measured event!) the γ-ray polarization is very large. A satisfactory theory of GRBs should naturally and very simply explain, among others, all these facts. We show that the "cannonball" (CB) model does it. In the CB model the process leading to the ejection of highly relativistic jetted CBs in core-collapse supernova (SN) explosions is akin to the one observed in quasars and microquasars. The prompt γ-ray emission -the GRB-is explained extremely well by inverse Compton scattering of light in the near environment of the SN by the electrons in the CBs' plasma. We have previously shown that the CB-model's description of GRB afterglows as synchrotron radiation from ambient electrons -swept in and accelerated within the CBs-is also simple, universal and very successful. The only obstacle still separating the CB model from a complete theory of GRBs is the theoretical understanding of the CBs' ejection mechanism in SN explosions. most dense star-burst regions, was found to be insufficient to explain the γ-ray fluence of the most powerful GRBs, such as GRB 990213. We shall see that, in the CB model, this problem -the dearth of "target" light-does not arise.In the CB model, long-duration GRBs are made by core-collapse supernovae (SNe). As we asserted in Dar & De Rújula (2000a) "the light from the SN shell is Compton upscattered to MeV energies, but its contribution to a GRB is sub-dominant". That assertion is correct: the light from the SN shell is too underluminous and too radially directed to generate GRBs of the observed fluence and individual-photon energy. With our collaborator Shlomo Dado, we have developed a very complete, simple and -we contend-extremely successful analysis of GRB AGs (Dado, Dar & De Rújula 2002a,b,c, 2003a. This thorough analysis has taught us that there should be another, much more intense and more isotropic, source of scattered light: the SN's "glory". The glory is the "echo" (or ambient) light from the SN, permeating the "wind-fed" circumburst density profile, previously ionized by the early extreme UV flash accompanying a SN explosion, or by the enhanced UV emission that precedes it. In Sections 2 and 3 we summarize the observations of pre-SN winds, early SN luminosities, and the UV flashes of SNe, to obtain the reference values of the v...

show abstract

Section: Appendix Ii: Ags In the Fb Models And In The Cb Modelmentioning

confidence: 91%

Towards a complete theory of gamma-ray bursts

2004

View full text Add to dashboard Cite

show abstract

“…2. When S is a linear subspace of A n (F) and the field F satisfies |F| ≥ n + 1, Theorem 1.2 is the case p = 2 of Theorem 2.1 in [3] that gives a lower bound on the maximal rank of a p-vector in a linear subspace S of the p-th exterior power p F n in terms of the weak matching number of a certain p-uniform hypergraph associated to S. It would be interesting to determine whether this result and its consequences remain true over arbitrary fields.…”

Section: Discussionmentioning

confidence: 99%

Maximal rank in matrix spaces via graph matchings

Meshulam

2017

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

Let M n (F) be the space of n×n matrices over a field F. A matrix A = A(i, j) n i,j=1 ∈ M n (F) is weakly symmetric if A(i, j) = 0 iff A(j, i) = 0 holds for all i, j. A matrix is alternating if it is skew-symmetric with zero diagonal. Let W n (F) and A n (F) denote respectively the set of weakly symmetric matrices and the space of alternating matricesFor a translate S of a linear space B ⊂ W n (F) let G S be the graph with loops on the vertex set [n] with edge set E S = {q(B) : 0 = B ∈ B}. A subset M ⊂ E S is a matching if e ∩ e ′ = ∅ for all e = e ′ ∈ M . Let µ(G S ) = max e∈M |e| where M ranges over all matchings M ⊂ E S . Let ρ(S) denote the maximal rank of a matrix in S. It is shown that if S is a translate of a linear space contained in W n (F) and |F| ≥ 3 then ρ(S) ≥ µ(G S ). The restriction on F can be removed if S is an affine subspace of A n (F). Applications include simple proofs of upper bounds on the dimension of affine subspaces of symmetric and alternating matrices of bounded rank. 2000 MSC: 05C50; 47L05

show abstract

“…A simple connection between k-spaces and graph matchings [14] provides an alternative proof of Flanders' result which is valid over arbitrary fields. Extensions of this combinatorial approach to bounded rank subspaces of (skew-)symmetric matrices, Toeplitz matrices and alternating tensors are given in [4,8,10,15].…”

Section: Introduction Let Fmentioning

confidence: 99%

Spaces of Singular Matrices and Matroid Parity

Gelbord

Meshulam

2002

European Journal of Combinatorics

Self Cite

View full text Add to dashboard Cite

Let V be a linear space of even dimension n over a field F of characteristic 0. A subspace W ⊂ ∧ 2 V is maximal singular if rank(w) ≤ n − 1 for all w ∈ W and any W W ⊂ ∧ 2 V contains a nonsingular matrix.It is shown that if W ⊂ ∧ 2 V is a maximal singular subspace which is generated by decomposable elements then dim W ≥ 3n 2 − 3 and that this bound is sharp. The main tool in the proof is the Lovász Matroid Parity Theorem.

show abstract

Spaces ofp-vectors of bounded rank

Cited by 5 publications

References 10 publications

Towards a complete theory of gamma-ray bursts

Towards a complete theory of gamma-ray bursts

Maximal rank in matrix spaces via graph matchings

Spaces of Singular Matrices and Matroid Parity

Contact Info

Product

Resources

About