Room squares with super-simple property

AbstractFrames have been defined as a certain type of generalization of Room square. Frames have proven useful in the construction of Room squares, in particular, skew Room squares.We generalize the definition of frame and consider the construction of Room squares and skew Room squares using these more general frames.We are able to construct skew Room squares of three previously unknown sides, namely 93, 159, and 237. This reduces the number of unknown sides to four: 69, 87, 95 and 123. Also, using this construction, we are able to give a short proof of the existence of all skew Room squares of (odd) sides exceeding 123.Finally, this frame construction is useful for constructing Room squares with subsquares. We can also construct Room squares “missing” subsquares of sides 3 and 5. The “missing” subsquares of sides 3 and 5 do not exist, so these incomplete Room squares cannot be completed to Room squares.

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Room Squares

Lecture Notes in Mathematics - Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices ◽

10.1007/bfb0069909 ◽

1972 ◽

pp. 30-121 ◽

Cited By ~ 3

Author(s):

W. D. Wallis

Keyword(s):

Room Squares

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Positions in Room squares

Combinatorics ◽

10.1017/cbo9780511662072.005 ◽

1974 ◽

pp. 23-26

Author(s):

R. L. Constable

Keyword(s):

Room Squares

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The Testing of Adhesives for Timber

Journal of the Royal Aeronautical Society ◽

10.1017/s0368393100135126 ◽

1929 ◽

Vol 33 (218) ◽

pp. 91-128

Author(s):

WM. D. Douglas ◽

C. B. Pettifor

Keyword(s):

Present State ◽

Test Piece ◽

High Stress ◽

Research Committee ◽

High Quality ◽

Scientific Societies ◽

Aircraft Structures ◽

Simple Property ◽

Timber Joints ◽

Control Variation

Mr. Douglas : From very early days adhesives have played an important part in the joiners’ trade. The necessity for consistent high quality in glues and cements became acute when they were used under conditions of high stress in aircraft structures. It is natural, therefore, to find that aeronautical interests have been largely represented on research committees which have investigated the use of adhesives. At the end of 1919 the Adhesives Research Committee of the Department of Scientific and Industrial Research was formed to continue the work of the Adhesives Committee of the Conjoint Board of Scientific Societies. It was realised that one of the obstacles to the study of adhesives was the unsatisfactory nature of the tests which determine their strength in timber joints. Many properties of glue (such as viscosity, jelly strength, etc.) have been used to control variation during manufacture or as an indication of probable relative strengths in timber joints, but it has never been generallyadmitted that any simple property of the glue itself can be accepted as representative of the ability of that glue to effect a joint with timber. In the present state of our knowledge, therefore, it appears to be necessary to make final appeal to the timber glue test piece.

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Rank, trace, eigenvalues and norms of a structured matrix

Asian-European Journal of Mathematics ◽

10.1142/s179355712040015x ◽

2019 ◽

Vol 12 (06) ◽

pp. 2040015

Author(s):

Ahmet İpek

Keyword(s):

Real Sequence ◽

Simple Property ◽

Structured Matrix ◽

The Matrix ◽

Matrix Formula ◽

Appl Math ◽

Real Matrices

The paper deals with rank, trace, eigenvalues and norms of the matrix [Formula: see text], where [Formula: see text] are ith components of any real sequence [Formula: see text]. A result in this paper is that the Euclidean and spectral norms of the matrix [Formula: see text] is [Formula: see text]. This is a generalization of the main result by Solak [Appl. Math. Comput. 232 (2014) 919–921], with the proof based on a simple property of norms of real matrices.

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