On the Existence of Positive Solutions of Quasilinear Elliptic Boundary Value Problems

AbstractWe establish the existence of positive solutions to a class of quasilinear anisotropic problems which have either sublinear or superlinear nonlinearity. With a, b nonnegative constants and α, β positive constants, one example is If b−a<1 (sublinear case), then for each λ∈[0,∞), (1) has a solution. On the other hand, if b−a>1 (superlinear case), then there exists a λ*>0 such that for 0⩽λ<λ*, (1) has at least one solution, and for λ>λ* no solution exists

A bacterial acetyltransferase triggers immunity in Arabidopsis thaliana independent of hypersensitive response

Type-III secreted effectors (T3Es) play critical roles during bacterial pathogenesis in plants. Plant recognition of certain T3Es can trigger defence, often accompanied by macroscopic cell death, termed the hypersensitive response (HR). Economically important species of kiwifruit are susceptible to Pseudomonas syringae pv. actinidiae (Psa), the causal agent of kiwifruit bacterial canker. Although Psa is non-pathogenic in Arabidopsis thaliana, we observed that a T3E, HopZ5 that is unique to a global outbreak clade of Psa, triggers HR and defence in Arabidopsis accession Ct-1. Ws-2 and Col-0 accessions are unable to produce an HR in response to Pseudomonas-delivered HopZ5. While Ws-2 is susceptible to virulent bacterial strain Pseudomonas syringae pv. tomato DC3000 carrying HopZ5, Col-0 is resistant despite the lack of an HR. We show that HopZ5, like other members of the YopJ superfamily of acetyltransferases that it belongs to, autoacetylates lysine residues. Through comparisons to other family members, we identified an acetyltransferase catalytic activity and demonstrate its requirement for triggering defence in Arabidopsis and Nicotiana species. Collectively, data herein indicate that HopZ5 is a plasma membrane-localized acetyltransferase with autoacetylation activity required for avirulence. ? 2017 The Author(s).115Ysciescopu

Sums of hermitian squares and the BMV conjecture

Recently Lieb and Seiringer showed that the Bessis-Moussa-Villani conjecture from quantum physics can be restated in the following purely algebraic way: The sum of all words in two positive semidefinite matrices where the number of each of the two letters is fixed is always a matrix with nonnegative trace. We show that this statement holds if the words are of length at most 13. This has previously been known only up to length 7. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming. As a by-product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators.Comment: 21 pages; minor changes; a companion Mathematica notebook is now available in the source fil

Robustness of quantum Markov chains

If the conditional information of a classical probability distribution of three random variables is zero, then it obeys a Markov chain condition. If the conditional information is close to zero, then it is known that the distance (minimum relative entropy) of the distribution to the nearest Markov chain distribution is precisely the conditional information. We prove here that this simple situation does not obtain for quantum conditional information. We show that for tri-partite quantum states the quantum conditional information is always a lower bound for the minimum relative entropy distance to a quantum Markov chain state, but the distance can be much greater; indeed the two quantities can be of different asymptotic order and may even differ by a dimensional factor.Comment: 14 pages, no figures; not for the feeble-minde

Local threshold field for dendritic instability in superconducting MgB2 films

Using magneto-optical imaging the phenomenon of dendritic flux penetration in superconducting films was studied. Flux dendrites were abruptly formed in a 300 nm thick film of MgB2 by applying a perpendicular magnetic field. Detailed measurements of flux density distributions show that there exists a local threshold field controlling the nucleation and termination of the dendritic growth. At 4 K the local threshold field is close to 12 mT in this sample, where the critical current density is 10^7 A/cm^2. The dendritic instability in thin films is believed to be of thermo-magnetic origin, but the existence of a local threshold field, and its small value are features that distinctly contrast the thermo-magnetic instability (flux jumps) in bulk superconductors.Comment: 6 pages, 6 figures, submitted to Phys. Rev.

Quantum capacity under adversarial quantum noise: arbitrarily varying quantum channels

We investigate entanglement transmission over an unknown channel in the presence of a third party (called the adversary), which is enabled to choose the channel from a given set of memoryless but non-stationary channels without informing the legitimate sender and receiver about the particular choice that he made. This channel model is called arbitrarily varying quantum channel (AVQC). We derive a quantum version of Ahlswede's dichotomy for classical arbitrarily varying channels. This includes a regularized formula for the common randomness-assisted capacity for entanglement transmission of an AVQC. Quite surprisingly and in contrast to the classical analog of the problem involving the maximal and average error probability, we find that the capacity for entanglement transmission of an AVQC always equals its strong subspace transmission capacity. These results are accompanied by different notions of symmetrizability (zero-capacity conditions) as well as by conditions for an AVQC to have a capacity described by a single-letter formula. In he final part of the paper the capacity of the erasure-AVQC is computed and some light shed on the connection between AVQCs and zero-error capacities. Additionally, we show by entirely elementary and operational arguments motivated by the theory of AVQCs that the quantum, classical, and entanglement-assisted zero-error capacities of quantum channels are generically zero and are discontinuous at every positivity point.Comment: 49 pages, no figures, final version of our papers arXiv:1010.0418v2 and arXiv:1010.0418. Published "Online First" in Communications in Mathematical Physics, 201

Multiplicativity of completely bounded p-norms implies a new additivity result

We prove additivity of the minimal conditional entropy associated with a quantum channel Phi, represented by a completely positive (CP), trace-preserving map, when the infimum of S(gamma_{12}) - S(gamma_1) is restricted to states of the form gamma_{12} = (I \ot Phi)(| psi >< psi |). We show that this follows from multiplicativity of the completely bounded norm of Phi considered as a map from L_1 -> L_p for L_p spaces defined by the Schatten p-norm on matrices; we also give an independent proof based on entropy inequalities. Several related multiplicativity results are discussed and proved. In particular, we show that both the usual L_1 -> L_p norm of a CP map and the corresponding completely bounded norm are achieved for positive semi-definite matrices. Physical interpretations are considered, and a new proof of strong subadditivity is presented.Comment: Final version for Commun. Math. Physics. Section 5.2 of previous version deleted in view of the results in quant-ph/0601071 Other changes mino

Critical behavior of the frustrated antiferromagnetic six-state clock model on a triangular lattice

We study the anti-ferromagnetic six-state clock model with nearest neighbor interactions on a triangular lattice with extensive Monte-Carlo simulations. We find clear indications of two phase transitions at two different temperatures: Below $T_I$ a chirality order sets in and by a thorough finite size scaling analysis of the specific heat and the chirality correlation length we show that this transition is in the Ising universality class (with a non-vanishing chirality order parameter below $T_I$). At $T_{KT}(<T_I)$ the spin-spin correlation length as well as the spin susceptibility diverges according to a Kosterlitz-Thouless (KT) form and spin correlations decay algebraically below $T_{KT}$. We compare our results to recent x-ray diffraction experiments on the orientational ordering of CF$_3$Br monolayers physisorbed on graphite. We argue that the six-state clock model describes the universal feature of the phase transition in the experimental system and that the orientational ordering belongs to the KT universality class.Comment: 8 pages, 9 figure

Relations for certain symmetric norms and anti-norms before and after partial trace

Changes of some unitarily invariant norms and anti-norms under the operation of partial trace are examined. The norms considered form a two-parametric family, including both the Ky Fan and Schatten norms as particular cases. The obtained results concern operators acting on the tensor product of two finite-dimensional Hilbert spaces. For any such operator, we obtain upper bounds on norms of its partial trace in terms of the corresponding dimensionality and norms of this operator. Similar inequalities, but in the opposite direction, are obtained for certain anti-norms of positive matrices. Through the Stinespring representation, the results are put in the context of trace-preserving completely positive maps. We also derive inequalities between the unified entropies of a composite quantum system and one of its subsystems, where traced-out dimensionality is involved as well.Comment: 11 pages, no figures. A typo error in Eq. (5.15) is corrected. Minor improvements. J. Stat. Phys. (in press