scholarly journals The Isoperimetric Number of the Incidence Graph of $PG(n,q)$

10.37236/6980 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Andrew Elvey Price ◽  
Muhammad Adib Surani ◽  
Sanming Zhou
Keyword(s):  
Projective Space ◽  
Prime Power ◽  
Incidence Graph ◽  
Order Of Magnitude ◽  
Isoperimetric Number

Let $\Gamma_{n,q}$ be the point-hyperplane incidence graph of the projective space $\operatorname{PG}(n,q)$, where $n \ge 2$ is an integer and $q$ a prime power. We determine the order of magnitude of $1-i_V(\Gamma_{n,q})$, where $i_V(\Gamma_{n,q})$ is the vertex-isoperimetric number of $\Gamma_{n,q}$. We also obtain the exact values of $i_V(\Gamma_{2,q})$ and the related incidence-free number of $\Gamma_{2,q}$ for $q \le 16$.

scholarly journals A geometrical representation theory for orthogonal arrays

1994 ◽  
Vol 49 (2) ◽  
pp. 311-324 ◽  
Author(s):  
David G. Glynn
Keyword(s):  
Representation Theory ◽  
Projective Space ◽  
Orthogonal Array ◽  
Prime Power ◽  
Infinite Order ◽  
Orthogonal Arrays ◽  
Geometrical Representation

Every orthogonal array of strength s and of prime-power (or perhaps infinite) order q, has a well-defined collection of ranks r. Having rank r means that it can be constructed as a cone cut by qs hyperplanes in projective space of dimension r over a field of order q.

scholarly journals Upper bounds on the length function for covering codes with covering radius $ R $ and codimension $ tR+1 $

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Alexander A. Davydov ◽  
Stefano Marcugini ◽  
Fernanda Pambianco
Keyword(s):  
Projective Space ◽  
Linear Code ◽  
Prime Power ◽  
Upper Bounds ◽  
Covering Radius ◽  
Length Function ◽  
Covering Codes ◽  
New Construction ◽  
One To One ◽  
Better Than

<p style='text-indent:20px;'>The length function <inline-formula><tex-math id="M3">\begin{document}$ \ell_q(r,R) $\end{document}</tex-math></inline-formula> is the smallest length of a <inline-formula><tex-math id="M4">\begin{document}$ q $\end{document}</tex-math></inline-formula>-ary linear code with codimension (redundancy) <inline-formula><tex-math id="M5">\begin{document}$ r $\end{document}</tex-math></inline-formula> and covering radius <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula>. In this work, new upper bounds on <inline-formula><tex-math id="M7">\begin{document}$ \ell_q(tR+1,R) $\end{document}</tex-math></inline-formula> are obtained in the following forms:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{split} &amp;(a)\; \ell_q(r,R)\le cq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &amp;\phantom{(a)\; } q\;{\rm{ is \;an\; arbitrary \;prime\; power}},\; c{\rm{ \;is\; independent \;of\; }}q. \end{split} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \begin{split} &amp;(b)\; \ell_q(r,R)&lt; 3.43Rq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &amp;\phantom{(b)\; } q\;{\rm{ is \;an\; arbitrary\; prime \;power}},\; q\;{\rm{ is \;large\; enough}}. \end{split} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>In the literature, for <inline-formula><tex-math id="M8">\begin{document}$ q = (q')^R $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M9">\begin{document}$ q' $\end{document}</tex-math></inline-formula> a prime power, smaller upper bounds are known; however, when <inline-formula><tex-math id="M10">\begin{document}$ q $\end{document}</tex-math></inline-formula> is an arbitrary prime power, the bounds of this paper are better than the known ones.</p><p style='text-indent:20px;'>For <inline-formula><tex-math id="M11">\begin{document}$ t = 1 $\end{document}</tex-math></inline-formula>, we use a one-to-one correspondence between <inline-formula><tex-math id="M12">\begin{document}$ [n,n-(R+1)]_qR $\end{document}</tex-math></inline-formula> codes and <inline-formula><tex-math id="M13">\begin{document}$ (R-1) $\end{document}</tex-math></inline-formula>-saturating <inline-formula><tex-math id="M14">\begin{document}$ n $\end{document}</tex-math></inline-formula>-sets in the projective space <inline-formula><tex-math id="M15">\begin{document}$ \mathrm{PG}(R,q) $\end{document}</tex-math></inline-formula>. A new construction of such saturating sets providing sets of small size is proposed. Then the <inline-formula><tex-math id="M16">\begin{document}$ [n,n-(R+1)]_qR $\end{document}</tex-math></inline-formula> codes, obtained by geometrical methods, are taken as the starting ones in the lift-constructions (so-called "<inline-formula><tex-math id="M17">\begin{document}$ q^m $\end{document}</tex-math></inline-formula>-concatenating constructions") for covering codes to obtain infinite families of codes with growing codimension <inline-formula><tex-math id="M18">\begin{document}$ r = tR+1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M19">\begin{document}$ t\ge1 $\end{document}</tex-math></inline-formula>.</p>

scholarly journals TETRAVALENT s-TRANSITIVE GRAPHS OF ORDER TWICE A PRIME POWER

2010 ◽  
Vol 88 (2) ◽  
pp. 277-288 ◽  
Author(s):  
JIN-XIN ZHOU ◽  
YAN-QUAN FENG
Keyword(s):  
Automorphism Group ◽  
Bipartite Graph ◽  
Projective Geometry ◽  
Prime Power ◽  
Complete Bipartite Graph ◽  
Three Dimensional ◽  
Transitive Graph ◽  
Incidence Graph ◽  
Normal Cover ◽  
Transitive Graphs

AbstractA graph is s-transitive if its automorphism group acts transitively on s-arcs but not on (s+1)-arcs in the graph. Let X be a connected tetravalent s-transitive graph of order twice a prime power. In this paper it is shown that s=1,2,3 or 4. Furthermore, if s=2, then X is a normal cover of one of the following graphs: the 4-cube, the complete graph of order 5, the complete bipartite graph K5,5 minus a 1-factor, or K7,7 minus a point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,2); if s=3, then X is a normal cover of the complete bipartite graph of order 4; if s=4, then X is a normal cover of the point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,3). As an application, we classify the tetravalent s-transitive graphs of order 2p2 for prime p.

scholarly journals Minimal Multiple Blocking Sets

10.37236/7810 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Anurag Bishnoi ◽  
Sam Mattheus ◽  
Jeroen Schillewaert
Keyword(s):  
Upper Bound ◽  
Prime Power ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Blocking Set ◽  
Blocking Sets ◽  
Incidence Graph ◽  
Finite Projective Spaces ◽  
Power Equality ◽  
Minimal Blocking

We prove that a minimal $t$-fold blocking set in a finite projective plane of order $n$ has cardinality at most \[\frac{1}{2} n\sqrt{4tn - (3t + 1)(t - 1)} + \frac{1}{2} (t - 1)n + t.\] This is the first general upper bound on the size of minimal $t$-fold blocking sets in finite projective planes and it generalizes the classical result of Bruen and Thas on minimal blocking sets. From the proof it directly follows that if equality occurs in this bound then every line intersects the blocking set $S$ in either $t$ points or $\frac{1}{2}(\sqrt{4tn  - (3t + 1)(t - 1)}  + t - 1) + 1$ points. We use this to show that for $n$ a prime power, equality can occur in our bound in exactly one of the following three cases: (a) $t = 1$, $n$ is a square and $S$ is a unital; (b) $t = n - \sqrt{n}$, $n$ is a square and $S$ is the complement of a Baer subplane; (c) $t = n$ and $S$ is equal to the set of all points except one. For a square prime power $q$ and $t \leq \sqrt{q} + 1$, we give a construction of a minimal $t$-fold blocking set $S$ in $\mathrm{PG}(2,q)$ with $|S| = q\sqrt{q} + 1 + (t - 1)(q - \sqrt{q} + 1)$. Furthermore, we obtain an upper bound on the size of minimal blocking sets in symmetric $2$-designs and use it to give new proofs of other known results regarding tangency sets in higher dimensional finite projective spaces. We also discuss further generalizations of our bound. In our proofs we use an incidence bound on combinatorial designs which follows from applying the expander mixing lemma to the incidence graph of these designs.

scholarly journals On Baer cones in $$\mathrm {PG}(3, q)$$

2021 ◽  
Vol 112 (3) ◽  
Author(s):  
Vito Napolitano
Keyword(s):  
Projective Space ◽  
Prime Power ◽  
Intersection Numbers ◽  
Link Type ◽  
Similar Characterization

AbstractRecently, in Innamorati and Zuanni (J. Geom 111:45, 2020. 10.1007/s00022-020-00557-0) the authors give a characterization of a Baer cone of $$\mathrm {PG}(3, q^2)$$ PG ( 3 , q 2 ) , q a prime power, as a subset of points of the projective space intersected by any line in at least one point and by every plane in $$q^2+1$$ q 2 + 1 , $$q^2+q+1$$ q 2 + q + 1 or $$q^3+q^2+1$$ q 3 + q 2 + 1 points. In this paper, we show that a similar characterization holds even without assuming that the order of the projective space is a square, and weakening the assumptions on the three intersection numbers with respect to the planes.

A controller for safe, convenient operation of LaB6 emitters on electron-beam instruments

1983 ◽  
Vol 41 ◽  
pp. 408-409
Author(s):  
W. J. Abramson ◽  
H. W. Estry ◽  
L. F. Allard
Keyword(s):  
Electron Beam ◽  
Control System ◽  
Thermal Shock ◽  
Appropriate Care ◽  
Electron Guns ◽  
Tungsten Filaments ◽  
Order Of Magnitude ◽  
Main Components

LaB6 emitters are becoming increasingly popular as direct replacements for tungsten filaments in the electron guns of modern electron-beam instruments. These emitters offer order of magnitude increases in beam brightness, and, with appropriate care in operation, a corresponding increase in source lifetime. They are, however, an order of magnitude more expensive, and may be easily damaged (by improper vacuum conditions and thermal shock) during saturation/desaturation operations. These operations typically require several minutes of an operator's attention, which becomes tedious and subject to error, particularly since the emitter must be cooled during sample exchanges to minimize damage from random vacuum excursions. We have designed a control system for LaBg emitters which relieves the operator of the necessity for manually controlling the emitter power, minimizes the danger of accidental improper operation, and makes the use of these emitters routine on multi-user instruments.Figure 1 is a block schematic of the main components of the control system, and Figure 2 shows the control box.

Novel magneto-optical recording materials: Bi-substituted garnet films

1991 ◽  
Vol 49 ◽  
pp. 776-777
Author(s):  
Takao Suzuki ◽  
Hossein Nuri
Keyword(s):  
Grain Size ◽  
Amorphous Film ◽  
Nitrogen Atmosphere ◽  
Optical Recording ◽  
Garnet Film ◽  
Media Noise ◽  
Garnet Films ◽  
Order Of Magnitude ◽  
A New Technique ◽  
A Current

For future high density magneto-optical recording materials, a Bi-substituted garnet film ((BiDy)3(FeGa)5O12) is an attractive candidate since it has strong magneto-optic effect at short wavelengths less than 600 nm. The signal in read back performance at 500 nm using a garnet film can be an order of magnitude higher than a current rare earth-transition metal amorphous film. However, the granularity and surface roughness of such crystalline garnet films are the key to control for minimizing media noise.We have demonstrated a new technique to fabricate a garnet film which has much smaller grain size and smoother surfaces than those annealed in a conventional oven. This method employs a high ramp-up rate annealing (Γ = 50 ~ 100 C/s) in nitrogen atmosphere. Fig.1 shows a typical microstruture of a Bi-susbtituted garnet film deposited by r.f. sputtering and then subsequently crystallized by a rapid thermal annealing technique at Γ = 50 C/s at 650 °C for 2 min. The structure is a single phase of garnet, and a grain size is about 300A.

Atomic-level imaging of the surface structure of Ag2O

1984 ◽  
Vol 42 ◽  
pp. 656-657
Author(s):  
William Krakow
Keyword(s):  
High Vacuum ◽  
Atomic Level ◽  
Ultra High Vacuum ◽  
Research Groups ◽  
Resolution Electron ◽  
Surface Reconstructions ◽  
Order Of Magnitude ◽  
High Resolution Electron Microscope ◽  
Saturated Structure ◽  
Transmission And Reflection

In recent years electron microscopy has been used to image surfaces in both the transmission and reflection modes by many research groups. Some of this work has been performed under ultra high vacuum conditions (UHV) and apparent surface reconstructions observed. The level of resolution generally has been at least an order of magnitude worse than is necessary to visualize atoms directly and therefore the detailed atomic rearrangements of the surface are not known. The present author has achieved atomic level resolution under normal vacuum conditions of various Au surfaces. Unfortunately these samples were exposed to atmosphere and could not be cleaned in a standard high resolution electron microscope. The result obtained surfaces which were impurity stabilized and reveal the bulk lattice (1x1) type surface structures also encountered by other surface physics techniques under impure or overlayer contaminant conditions. It was therefore decided to study a system where exposure to air was unimportant by using a oxygen saturated structure, Ag2O, and seeking to find surface reconstructions, which will now be described.

Near-field scanning optical microscopy

1986 ◽  
Vol 44 ◽  
pp. 642-643
Author(s):  
E. Betzig ◽  
A. Harootunian ◽  
M. Isaacson ◽  
A. Lewis
Keyword(s):  
Near Field ◽  
Optical Microscope ◽  
Visible Radiation ◽  
Field Methods ◽  
Optical Elements ◽  
Optical Region ◽  
Order Of Magnitude ◽  
Nondestructive Imaging ◽  
Scanning Optical Microscopy ◽  
Near Field Scanning

In general, conventional methods of optical imaging are limited in spatial resolution by either the wavelength of the radiation used or by the aberrations of the optical elements. This is true whether one uses a scanning probe or a fixed beam method. The reason for the wavelength limit of resolution is due to the far field methods of producing or detecting the radiation. If one resorts to restricting our probes to the near field optical region, then the possibility exists of obtaining spatial resolutions more than an order of magnitude smaller than the optical wavelength of the radiation used. In this paper, we will describe the principles underlying such "near field" imaging and present some preliminary results from a near field scanning optical microscope (NS0M) that uses visible radiation and is capable of resolutions comparable to an SEM. The advantage of such a technique is the possibility of completely nondestructive imaging in air at spatial resolutions of about 50nm.

Detection, Averaging, and 3D Reconstruction of Biological Specimens on Hypercubes (Transputer-based) Computers

1990 ◽  
Vol 48 (1) ◽  
pp. 454-455
Author(s):  
Jose-Maria Carazo ◽  
I. Benavides ◽  
S. Marco ◽  
J.L. Carrascosa ◽  
E.L. Zapata
Keyword(s):  
3D Reconstruction ◽  
3D Structure ◽  
Three Dimensional ◽  
Software Tools ◽  
Mapping Method ◽  
Computer Architectures ◽  
Parallel Computer ◽  
Reconstruction Process ◽  
Computing Power ◽  
Order Of Magnitude

Obtaining the three-dimensional (3D) structure of negatively stained biological specimens at a resolution of, typically, 2 - 4 nm is becoming a relatively common practice in an increasing number of laboratories. A combination of new conceptual approaches, new software tools, and faster computers have made this situation possible. However, all these 3D reconstruction processes are quite computer intensive, and the middle term future is full of suggestions entailing an even greater need of computing power. Up to now all published 3D reconstructions in this field have been performed on conventional (sequential) computers, but it is a fact that new parallel computer architectures represent the potential of order-of-magnitude increases in computing power and should, therefore, be considered for their possible application in the most computing intensive tasks.We have studied both shared-memory-based computer architectures, like the BBN Butterfly, and local-memory-based architectures, mainly hypercubes implemented on transputers, where we have used the algorithmic mapping method proposed by Zapata el at. In this work we have developed the basic software tools needed to obtain a 3D reconstruction from non-crystalline specimens (“single particles”) using the so-called Random Conical Tilt Series Method. We start from a pair of images presenting the same field, first tilted (by ≃55°) and then untilted. It is then assumed that we can supply the system with the image of the particle we are looking for (ideally, a 2D average from a previous study) and with a matrix describing the geometrical relationships between the tilted and untilted fields (this step is now accomplished by interactively marking a few pairs of corresponding features in the two fields). From here on the 3D reconstruction process may be run automatically.

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