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Submitted 1 March, 2017; originally announced March 2017.

Comments: 6 pages; comments welcome!

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[, , other ] math.CO

On Cartesian Products which Determine Few Distinct Distances

Abstract : Every set of points \mathcal{P} P determines Ω(|\mathcal{P}| / \log |\mathcal{P}|) distances. A close version of this was initially conjectured by Erdős in 1946 and rather recently proved by Guth and Katz. We show that when near this lower bound, a point set \mathcal{P} of the form A \times A must satisfy |A - A| \ll |A|^{2-\frac{2}{7}} \log^{\frac{1}{7}} |A| . This improves recent results… ▽ More Every set of points \mathcal{P} determines Ω(|\mathcal{P}| / \log |\mathcal{P}|) distances. A close version of this was initially conjectured by Erdős in 1946 and rather recently proved by Guth and Katz. We show that when near this lower bound, a point set \mathcal{P} of the form A \times A must satisfy |A - A| \ll |A|^{2-\frac{2}{7}} \log^{\frac{1}{7}} |A| . This improves recent results of Hanson and Roche-Newton. △ Less

Submitted 25 March, 2018; v1 submitted 19 December, 2016; originally announced December 2016.

arXiv:0807.1133 [, CafePress Martial Arts Female Red Black Flip Flops Funny Thong Sandals Beach Sandals Caribbean Blue vVy5d9fj
]
math.CO

Boole's formula as a consequence of Lagrange's Interpolating Polynomial theorem

Abstract : We present a slightly more general version of Boole's additive formula for factorials as a simple consequence of Lagrange's Interpolating Polynomial theorem. ▽ More We present a slightly more general version of Boole's additive formula for factorials as a simple consequence of Lagrange's Interpolating Polynomial theorem. △ Less

Submitted 7 July, 2008; originally announced July 2008.

Comments: 3 pages

MSC Class: 11-03; 01A60

Journal ref: Integers: Electronic Journal of Combinatorial Number Theory 8 (2008), A23

arXiv:0807.1131 [, Jellypop Womens Leo Engineer Boot Black Distress IntRma4
]
math.HO

Generalization of the Apollonius Circles

Authors: Cosmin Pohoata , Vladimir Zajic

Abstract : The three Apollonius circles of a triangle, each passing through a triangle vertex, the corresponding vertex of the cevian triangle of the incenter and the corresponding vertex of the circumcevian triangle of the symmedian point, are coaxal. Similarly defined three circles remain coaxal, when the circumcevian triangle is defined with respect to any point on the triangle circumconic through the i… ▽ More The three Apollonius circles of a triangle, each passing through a triangle vertex, the corresponding vertex of the cevian triangle of the incenter and the corresponding vertex of the circumcevian triangle of the symmedian point, are coaxal. Similarly defined three circles remain coaxal, when the circumcevian triangle is defined with respect to any point on the triangle circumconic through the incenter and symmedian point. Inversion in the incircle of the reference triangle carries these three coaxal circles into coaxal circles, each passing through a vertex of the inverted triangle and centered on the opposite sideline, at the intersection of the orthotransversal with respect to a point on the Euler line of the inverted triangle. A similar circumconic exists in a more general configuration, when the cevian triangle is defined with respect to an arbitrary point, passing through this arbitrary point and isogonal conjugate of its complement. △ Less

BILL JOHNSON
Leadership Series

On April 30, Rep. Johnson spoke at Union Local High School in Belmont County as part of his student Leadership Series, which is taking place at various high schools across Eastern and Southeastern Ohio.

National Automotive Chemical

Rep. Johnson toured and learned about the soap production process the company uses during a visit to National Automotive Chemical in Cambridge on March 9.

VFW Post 9571 70th Anniversary

Bill Johnson traveled to Ellsworth in Mahoning County on February 24 to celebrate the 70th anniversary of VFW 9571 Meander Post.

Barnesville Robotics Team

Bill Johnson presented a proclamation to the Barnesville Middle School Robotics Team on April 30. The students recently won a statewide competition.

Ironton Flood Gates

Rep. Bil Johnson in Ironton on February 21 checking out the city's flood gates. It takes 12 people 12 hours to get the gates up and ready.

Meigs Broadband Roundtable

Bill Johnson recently held a roundtable in Meigs County to discuss the important issue of rural broadband.

Dogs 4 Warriors

During a recent District Work Period, Rep. Johnson visited Dogs 4 Warriors, which is located just outside Bowerston. It is a non-profit organization that provides professionally trained service dogs at no cost to combat veterans.

Youngstown Delegation Visit

On January 17, Bill Johnson met with YSU President Tressel, Youngstown Mayor Brown, other elected officials, business community representatives, and other local leaders to discuss economic development opportunities that will benefit Youngstown and the entire Mahoning Valley.

Appreciation for Our Veterans

On September 20, Bill Johnson joined 5th and 6th graders at Bloom-Vernon Elementary School in South Webster in writing letters of appreciation to local veterans.

Noble County Chamber of Commerce

Bill Johnson recently met with members of the Noble County Chamber of Commerce in Caldwell to discuss local road and infrastructure issues.

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