Sign patterns of rational matrices with large rank II

It is known that, for any real m-by-n matrix A of rank n-2, there is a rational m-by-n matrix which has rank n-2 and sign pattern equal to that of A. We prove a more general result conjectured in the recent literature.

On the infinite Borwein product raised to a positive real power

In this paper, we study properties of the coefficients appearing in the q-series expansion of $$\prod _{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta $$ ∏ n ≥ 1 [ ( 1 - q n ) / ( 1 - q pn ) ] δ , the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power $$\delta $$ δ . We use the Hardy–Ramanujan–Rademacher circle method to give an asymptotic formula for the coefficients. For $$p=3$$ p = 3 we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $$\delta $$ δ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $$p=3$$ p = 3 case.

On the Eventual Exponential Positivity of Some Tree Sign Patterns

An n×n matrix A is called eventually exponentially positive (EEP) if etA=∑k=0∞tkAkk!>0 for all t≥t0, where t0≥0. A matrix whose entries belong to the set {+,−,0} is called a sign pattern. An n×n sign pattern A is called potentially eventually exponentially positive (PEEP) if there exists some real matrix realization A of A that is EEP. Characterizing the PEEP sign patterns is a longstanding open problem. In this article, A is called minimally potentially eventually exponentially positive (MPEEP), if A is PEEP and no proper subpattern of A is PEEP. Some preliminary results about MPEEP sign patterns and PEEP sign patterns are established. All MPEEP sign patterns of orders n≤3 are identified. For the n×n tridiagonal sign patterns Tn, we show that there exists exactly one MPEEP tridiagonal sign pattern Tno. Consequently, we classify all PEEP tridiagonal sign patterns as the superpatterns of Tno. We also classify all PEEP star sign patterns Sn and double star sign patterns DS(n,m) by identifying all the MPEEP star sign patterns and the MPEEP double star sign patterns, respectively.

Mandibular Fractures: Pre-Operative Panoramic Radiography and Duplication Sign Patterns

The purpose of this essay is to present the multiple patterns of the duplication sign at the mandibular fracture line/gap visualized at the panoramic radiography as two-line fracture gap or pseudocomminuted fracture. We retrospectively reviewed the orthopantomography of patients with mandible fractures and presented nine patients with 12 duplication signs (also known as lambda course fracture line). On panoramic radiographs the fracture line/gap with duplication sign is visualized as two-line cortical bone discontinuity with or without dislocation due to the fact that the fracture gap runs asymmetrically on the external and internal cortical plates of the jaw. Knowledge of duplication sign patterns, artifacts is also crucial for the precise diagnosis and choice of correct management strategy.

Near-field strain in DAS-based microseismic observation

Microseismic monitoring with surface or downhole geophone arrays has been commonly used in tracking subsurface deformation and fracture networks during hydraulic fracturing operations. Recently, the use of fiber-optic DAS technology has improved microseismic acquisition to a new level with unprecedentedly high spatial resolution and low cost. Deploying fiber-optic cables in horizontal boreholes allows very close observation of these micro-sized earthquakes and captures their full wavefield details. We show that DAS-based microseismic profiles present a seldomly reported near-field strain signal between the P- and S-wave arrivals. This near-field signal shows monotonically increasing (or decreasing) temporal variation, which resembles the previously reported near-field observations of large earthquakes. To understand the near-field strain behavior, we provide a mathematical expression of the analytic normal strain solution that reveals the near-field, intermediate-near-field, intermediate-far-field, and far-field components. Synthetic DAS strain records of hydraulic-fracture-induced microseismic events can be generated using this analytic solution with the Brune source model. The polarity sign patterns of the near-field and far-field terms in these synthetics are linked to the corresponding source mechanism’s radiation patterns. These polarity sign patterns are demonstrated to be sensitive to the source orientations by rotating the moment tensor in different directions. A field data example is compared to the synthetic result and a qualitative match is shown. The microseismic near-field signals detected by DAS have potential value in hydraulic fracture monitoring by providing a means to better constrain microseismic source parameters that characterize the source magnitude, source orientation, and temporal source evolution, and therefore better reflect the geomechanical response of the hydraulically fractured environment in the unconventional reservoirs.