Existence of an Antisymmetric Solution of a Boundary Value Problem with Antiperiodic Boundary Conditions

In this work, an application is made of a recent extension of the Leggett-Williams fixed point theorem, commonly referred to as an Avery type fixed point theorem, to a second order boundary value problem with antiperiodic boundary conditions. Under certain conditions and with the use of concavity, an antisymmetric solution to the boundary value problem is shown to exist. In conclusion, a non-trivial example is provided

Be Prepared: Unsuspecting Employers Are Vulnerable for Title VII Sexual Harassment Environment Claims

CONGRESS ENACTED TITLE VII of the Civil Rights Act of 19641 ( Tide VII ) to protect employees from discrimination in the workplace. 2 Over the past three decades, however, Title VII hostile work environment actions have expanded potential liability for employers to the point that the employers must take preventive measures or be susceptible to unexpected Title VII suits. An unprepared employer could be caught off guard in situations where, over the course of an employee\u27s tenure, the employee endures occasional pranks, or is exposed to occasional sexual epithets, or coarse or vulgar language. While each of these situations clearly exposes the employee to an unsavory work environment, individually they may not cross the threshold of an actionable Title VII claim. However, the court might consider this conduct collectively, and thus the employer may be liable for creating or maintaining a hostile work environment in violation of Title VII, even if no individual instance would have violated the Ac

A Difference Equation with Anti-Periodic Boundary Conditions

In this paper, we apply an extension of the Leggett-Williams fixed point theorem to the second order difference equation ∆2u(k)+f(u(k+1)) = 0, k ∈ {0, 1, . . . , N}, satisfying the anti-periodic boundary conditions u(0) + u(N + 2) = 0, ∆u(0) + ∆u(N +1) = 0. Two important results of this paper involve providing the Green’s function for −∆2u(k) = 0 satisfying u(0) + u(N + 2) = 0, ∆u(0) + ∆u(N + 1) = 0 and showing this Green’s function satisfies a concavity like property. An example is also given

Existence of an antisymmetric solution of a boundary value problem with antiperiodic boundary conditions

In this work, an application is made of a recent extension of the Leggett-Williams fixed point theorem, commonly referred to as an Avery type fixed point theorem, to a second order boundary value problem with antiperiodic boundary conditions. Under certain conditions and with the use of concavity, an antisymmetric solution to the boundary value problem is shown to exist. In conclusion, a non-trivial example is provided

Positive solutions of a singular fractional boundary value problem with a fractional boundary condition

For $\alpha\in(1,2]$, the singular fractional boundary value problem $$D^{\alpha}_{0^+}x+f\left(t,x,D^{\mu}_{0^+}x\right)=0,\quad 0\lt t\lt 1,$$ satisfying the boundary conditions $x(0)=D^{\beta}_{0^+}x(1)=0$, where $\beta\in(0,\alpha-1]$, $\mu\in(0,\alpha-1]$, and $D^{\alpha}_{0^+}$, $D^{\beta}_{0^+}$ and $D^{\mu}_{0^+}$ are Riemann-Liouville derivatives of order $\alpha$, $\beta$ and $\mu$ respectively, is considered. Here $f$ satisfies a local Carathéodory condition, and $f(t,x,y)$ may be singular at the value 0 in its space variable $x$. Using regularization and sequential techniques and Krasnosel\u27skii\u27s fixed point theorem, it is shown this boundary value problem has a positive solution. An example is given

Isolated sequences from the linked Myf-5 and MRF4 genes drive distinct patterns of muscle-specific expression in transgenic mice

In developing mouse embryos, MyoD family regulatory genes are expressed specifically in muscle precursors and mature myofibers. This pattern, taken together with the well-established ability of MyoD family members to convert a variety of cell types to skeletal muscle, suggests a significant role for these genes in regulating skeletal myogenesis. The possibility that expression of these genes may be causally associated with segregation of the myogenic lineage from other mesodermal derivatives, or with the subsequent maintenance of muscle phenotypes at later times, raises the issue of how MyoD family genes are themselves regulated during development. In this work, we have initiated studies to identify DNA sequences that govern Myf-5 and MRF4 (herculin, myf-6) transcription. Myf-5 is the first of the MyoD family to be expressed in the developing mouse embryo, while MRF4 is the most abundantly expressed myogenic factor in postnatal animals. In spite of their strikingly divergent patterns of expression, Myf-5 and MRF4 are tightly linked in the mouse genome; their translational start codons are only 8.5 kilobases apart. Here, the 5' flanking regions of the mouse Myf-5 and MRF4 genes were separately linked to a bacterial β-galactosidase (lacZ) gene, and these constructs were each used to produce several lines of transgenic mice. Transgene expression was monitored by X-gal staining of whole embryos and by in situ hybridization of embryo sections. For the Myf-5/lacZ lines, the most intense transgene expression was in the visceral arches and their craniofacial muscle derivatives, beginning at day 8.75 post coitum (p.c.). This correlates with endogenous Myf-5 expression in visceral arches. However, while Myf-5 is also expressed in somites starting at day 8 p.c., transgene expression in the trunk is not observed until day 12 p.c. Thus, the Myf-5/lacZ construct responds to early Myf-5 activators in the visceral arches but not in the somites, suggesting that myogenic determination in the nonsomitic head mesoderm may be under separate control from that of the somitic trunk mesoderm. MRF4/lacZ lines displayed an entirely different pattern from Myf-5. Transgene expression appeared in muscles starting at day 16.5 p.c. and became increasingly prominent at later times. However, an early wave of myotomal expression that is characteristic of the endogenous MRF4 was not recapitulated by the transgene

Where you live and who your neighbors are influences whether or not you think closing the income gap is important

Income inequality and its effects has become a pressing concern for politicians across the political spectrum. But what makes people sit up and take notice of the inequality that may surround them? In new research which focuses on income diversity in New York, Scott Minkoff and Jeffrey Lyons find that those who live in areas with a greater diversity of incomes are more likely to perceive that there is a larger income gap compared to those who live in areas which tend to be generally poor or generally wealthy

The Seas Are Changing: It’s Time to Use Ocean-Based Renewable Energy, the Public Trust Doctrine, And a Green Thumb to Protect Seas From our Changing Climate

Climate change is having significant impacts on ocean and coastal ecosystems and wildlife, with rising seas inundating wetlands and coastal estuaries, warming waters altering marine species habitat and breeding behavior, and ocean acidification weakening corals, mollusks, and marine species, leaving them vulnerable to disease and predation. Though the threat of climate change has been acknowledged by President Obama and in federal statutory language,5 to date the U.S. government*242 has yet to provide a comprehensive plan to address the detrimental impacts of warming lands and seas. In response to governmental inaction, this Article seeks to demonstrate how the public trust doctrine (PTD) can play a role in protecting ocean and coastal resources from climate change. More specifically, this Article proposes that both federal and state PTDs can help protect traditional trust values6 of commerce, navigation and fishing-in addition to modern trust values of protecting tidal wetlands, estuaries, and wildlife-through establishing ocean-based renewable energy (ORE) as a public trust value. In addition to elevating ORE to equal footing with traditional trust values, this Article calls for placing a “green thumb” on the scales of balancing competing trust values to explicitly guide courts and agencies alike to operate under a rebuttable presumption favoring ORE over other PTD values because of its ability to help reduce carbon and other greenhouse gas (GHG) emissions as a means to mitigate climate change. As Bob Dylan wrote and sang fifty years ago, the times are changin’ and waters are growing rapidly – today it is well past time to heed that warning and combat the rising levels of greenhouse gases, temperatures, seas, health care costs and storm damages by making maximum use of the clean, renewable energy available and waiting off our shores. Establishing a federal PTD and ORE values in all PTDs will enable governments to better fulfill their fiduciary obligations while empowering the doctrine to proactively protect trust resources – both traditional and modern – from the impacts of climate change

Scaffolding the Implementation of the Engineering Design Process within STEM Based Projects

The purpose of this research was to improve student understanding and use of the engineering design process by scaffolding instruction of implementation during STEM project-based learning. The study was conducted in a fifth-grade engineering class and a seventh and eighth grade technology class with a total of 79 participants. The researchers collected data through pre and post student questionnaires, student checklists, researcher observations and reflection journals. Findings indicated that scaffolding instruction improved student understanding and implementation of the engineering design process. Further research could indicate the effectiveness of teaching best practices within each step of the process and further understanding within STEM project-based learning activities