scholarly journals A Survey of the Hadamard Maximal Determinant Problem

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Patrick Browne ◽  
Ronan Egan ◽  
Fintan Hegarty ◽  
Padraig Ó Catháin
Keyword(s):  
Historical Development ◽  
Design Theory ◽  
Infinite Family ◽  
Upper Bounds ◽  
Quadratic Residues ◽  
Maximum Value ◽  
Gram Matrices ◽  
The Subject ◽  
First Time ◽  
Determinant Theory

In a celebrated paper of 1893, Hadamard established the maximal determinant theorem, which establishes an upper bound on the determinant of a matrix with complex entries of norm at most 1. His paper concludes with the suggestion that mathematicians study the maximum value of the determinant of an $n \times n$ matrix with entries in $\{ \pm 1\}$. This is the Hadamard maximal determinant problem. This survey provides complete proofs of the major results obtained thus far. We focus equally on upper bounds for the determinant (achieved largely via the study of the Gram matrices), and constructive lower bounds (achieved largely via quadratic residues in finite fields and concepts from design theory). To provide an impression of the historical development of the subject, we have attempted to modernise many of the original proofs, while maintaining the underlying ideas. Thus some of the proofs have the flavour of determinant theory, and some appear in print in English for the first time. We survey constructions of matrices in order $n \equiv 3 \mod 4$, giving asymptotic analysis which has not previously appeared in the literature. We prove that there exists an infinite family of matrices achieving at least 0.48 of the maximal determinant bound. Previously the best known constant for a result of this type was 0.34.

QUANTUM DESIGNS: FOUNDATIONS OF A NONCOMMUTATIVE DESIGN THEORY

2011 ◽  
Vol 09 (01) ◽  
pp. 445-507 ◽  
Author(s):  
GERHARD ZAUNER
Keyword(s):  
Design Theory ◽  
Numerical Solutions ◽  
Infinite Family ◽  
Positive Operator Valued Measures ◽  
Master's Thesis ◽  
The 1960S ◽  
Equiangular Lines ◽  
First Time ◽  
Projection Matrices ◽  
Special Case

This is a one-to-one translation of a German-written Ph.D. thesis from 1999. Quantum designs are sets of orthogonal projection matrices in finite(b)-dimensional Hilbert spaces. A fundamental differentiation is whether all projections have the same rank r, and furthermore the special case r = 1, which contains two important subclasses: Mutually unbiased bases (MUBs) were introduced prior to this thesis and solutions of b + 1 MUBs whenever b is a power of a prime were already given. Unaware of those papers, this concept was generalized here under the notation of regular affine quantum designs. Maximal solutions are given for the general case r ≥ 1, consisting of r(b2 - 1)/(b - r) so-called complete orthogonal classes whenever b is a power of a prime. For b = 6, an infinite family of MUB triples was constructed and it was — as already done in the author's master's thesis (1991) — conjectured that four MUBs do not exist in this dimension. Symmetric informationally complete positive operator-valued measures (SIC POVMs) in this paper are called regular quantum 2-designs with degree 1. The assigned vectors span b2 equiangular lines. These objects had been investigated since the 1960s, but only a few solutions were known in complex vector spaces. In this thesis further maximal analytic and numerical solutions were given (today a lot more solutions are known) and it was (probably for the first time) conjectured that solutions exist in any dimension b (generated by the Weyl–Heisenberg group and with a certain additional symmetry of order 3).

scholarly journals On Galileo’s Tallest Column

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Mariano Vázquez Espí ◽  
Jaime Cervera Bravo ◽  
Carlos Olmedo Rojas
Keyword(s):  
Cross Section ◽  
Structural Design ◽  
Design Theory ◽  
Shape Function ◽  
Affirmative Answer ◽  
Infinite Height ◽  
The Subject ◽  
Definition Of ◽  
First Time ◽  
Constant Maximum

The height at which an unloaded column will fail under its own weight was calculated for first time by Galileo for cylindrical columns. Galileo questioned himself if there exists a shape function for the cross section of the column with which it can attain a greater height than the cylindrical column. The problem is not solved since then, although the definition of the so named “constant maximum strength” solids seems to give an affirmative answer to Galileo’s question, in the form of shapes which seem to attain infinite height, even when loaded with a useful load at the top. The main contribution of this work is to show that Galileo’s problem is (i) an important problem for structural design theory of buildings and other structures, (ii) not solved by the time being in any sense, and (iii) an interesting problem for mathematicians involved in related but very different problems (as Euler’s tallest column). A contemporary formulation of the problem is included as a result of a research on the subject.

СОЦИАЛЬНАЯ КОНФЛИКТОЛОГИЯ: ТЕОРЕТИКО-ИСТОРИЧЕСКИЙ АСПЕКТ

2020 ◽  
Vol 15 (2) ◽  
pp. 68
Author(s):  
А. Н. Сухов
Keyword(s):  
Theory And Practice ◽  
Important Task ◽  
Social Conflicts ◽  
Psychological Approach ◽  
Structure Dynamics ◽  
History Of ◽  
The Subject ◽  
First Time

This given article reveals the topicality not only of destructive, but also of constructive, as well as hybrid conflicts. Practically it has been done for the first time. It also describes the history of the formation of both foreign and domestic social conflictology. At the same time, the chronology of the development of the latter is restored and presented objectively, in full, taking into account the contribution of those researchers who actually stood at its origins. The article deals with the essence of the socio-psychological approach to understanding conflicts. The subject of social conflictology includes the regularities of their occurrence and manifestation at various levels, spheres and conditions, including normal, complicated and extreme ones. Social conflictology includes the theory and practice of diagnosing, resolving, and resolving social conflicts. It analyzes the difficulties that occur in defining the concept, structure, dynamics, and classification of social conflicts. Therefore, it is no accident that the most important task is to create a full-fledged theory of social conflicts. Without this, it is impossible to talk about effective settlement and resolution of social conflicts. Social conflictology is an integral part of conflictology. There is still a lot of work to be done, both in theory and in application, for its complete design. At present, there is an urgent need to develop conflict-related competence not only of professionals, but also for various groups of the population.

Thermodynamics in Geochemistry

1993 ◽  
Author(s):  
Greg M. Anderson ◽  
David A. Crerar
Keyword(s):  
Teaching Experience ◽  
Graduate Level ◽  
Years Of Teaching Experience ◽  
The Subject ◽  
First Time

This textbook and reference outlines the fundamental principles of thermodynamics, emphasizing applications in geochemistry. The work is distinguished by its comprehensive, balanced coverage and its rigorous presentation. The authors bring years of teaching experience to the work, and have attempted to particularly address those areas where other texts on the subject have provided inadequate coverage. A thorough review of the necessary mathematics is presented early on, both as a refresher for those with a background in university calculus, and for the benefit of those coming to the subject for the first time. The text is written for students in advanced undergraduate or graduate-level geochemistry as well as for all researchers in this field.

Anisotropic Elasticity

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
Composite Materials ◽  
Elastic Constants ◽  
Piezoelectric Materials ◽  
Anisotropic Materials ◽  
Anisotropic Elasticity ◽  
Stress Singularities ◽  
Comprehensive Survey ◽  
The Subject ◽  
Key Topics ◽  
First Time

Anisotropic Elasticity offers for the first time a comprehensive survey of the analysis of anisotropic materials that can have up to twenty-one elastic constants. Focusing on the mathematically elegant and technically powerful Stroh formalism as a means to understanding the subject, the author tackles a broad range of key topics, including antiplane deformations, Green's functions, stress singularities in composite materials, elliptic inclusions, cracks, thermo-elasticity, and piezoelectric materials, among many others. Well written, theoretically rigorous, and practically oriented, the book will be welcomed by students and researchers alike.

scholarly journals Coefficients of a Comprehensive Subclass of Meromorphic Bi-Univalent Functions Associated with the Faber Polynomial Expansion

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 27
Author(s):  
Hari Mohan Srivastava ◽  
Ahmad Motamednezhad ◽  
Safa Salehian
Keyword(s):  
Unit Disk ◽  
Univalent Functions ◽  
Expansion Method ◽  
Upper Bounds ◽  
Polynomial Expansion ◽  
Open Unit ◽  
Open Unit Disk ◽  
Faber Polynomial ◽  
The Subject ◽  
Polynomial Expansion Method

In this paper, we introduce a new comprehensive subclass ΣB(λ,μ,β) of meromorphic bi-univalent functions in the open unit disk U. We also find the upper bounds for the initial Taylor-Maclaurin coefficients |b0|, |b1| and |b2| for functions in this comprehensive subclass. Moreover, we obtain estimates for the general coefficients |bn|(n≧1) for functions in the subclass ΣB(λ,μ,β) by making use of the Faber polynomial expansion method. The results presented in this paper would generalize and improve several recent works on the subject.

scholarly journals Per Ravenna trecentesca: nuove proposte per l’assetto architettonico di Santa Maria in Porto Fuori

2019 ◽  
Vol 82 (1) ◽  
pp. 3-31
Author(s):  
Fabio Massaccesi
Keyword(s):  
Fourteenth Century ◽  
Second World War ◽  
World War ◽  
The Subject ◽  
First Time ◽  
Santa Maria ◽  
Second World ◽  
The Church

Abstract This contribution intends to draw attention to one of the most significant monuments of medieval Ravenna: the church of Santa Maria in Porto Fuori, which was destroyed during the Second World War. Until now, scholars have focused on the pictorial cycle known through photographs and attributed to the painter Pietro da Rimini. However, the architecture of the building has not been the subject of systematic studies. For the first time, this essay reconstructs the fourteenth-century architectural structure of the church, the apse of which was rebuilt by 1314. The data that led to the virtual restitution of the choir and the related rood screen are the basis for new reflections on the accesses to the apse area, on the pilgrimage flows, and on the view of the frescoes.

Offshore Claims and Problems in the North Sea

1965 ◽  
Vol 59 (3) ◽  
pp. 505-522 ◽  
Author(s):  
Richard Young
Keyword(s):  
North Sea ◽  
Large Scale ◽  
Geneva Convention ◽  
Legal Principles ◽  
The North ◽  
Legal Problems ◽  
The North Sea ◽  
The Subject ◽  
First Time ◽  
Economic Considerations

The possible presence of very large petroleum and natural gas reserves in the area beneath the North Sea is currently the subject of intense investigation. If confirmed, as seems likely in at least some localities, this occurrence will raise legal problems of considerable interest and complexity. For the North Sea is not merely an oilfield covered by water: for centuries it has been one of the world's major fishery regions and the avenue to and from the world's busiest seaports. Thus all three of the present principal uses of the sea—fishing, navigation, and the exploitation of submarine resources—promise to meet for the first time on a large scale in an area where all are of major importance. The process of reconciling the various interests at stake will provide the first thoroughgoing test of the adequacy and acceptability of the general principles laid down in the 1958 Geneva Convention on the Continental Shelf and should add greatly to the practice and precedents available in this developing branch of the law. In the present article an attempt is made to review some of the geographical and economic considerations involved in the North Sea situation, to note some of the technical and legal developments that have already taken place, and to consider these elements in the light of the various interests and legal principles concerned.

The Squire Law Library at Cambridge: the Historical Development and Current Status of the International Law Collections

2017 ◽  
Vol 17 (2) ◽  
pp. 78-91 ◽  
Author(s):  
Lesley Dingle
Keyword(s):  
International Law ◽  
Historical Development ◽  
Detailed Account ◽  
Current Status ◽  
Public International Law ◽  
The Public ◽  
The Subject ◽  
Law Library

AbstractIn this paper Lesley Dingle provides a detailed account of the historical development of the public international law collections at the Squire Law Library in Cambridge. She explains the close involvement of the academic lawyers and the librarians, past and present, in developing an important collection which reflects the significance of the subject at Cambridge's Faculty of Law. Finally, she brings things up-to-date by detailing the extent of the electronic provision which benefits the modern scholar in this discipline.

Insanity and Divorce

1892 ◽  
Vol 38 (162) ◽  
pp. 378-382
Author(s):  
A. Wood Renton
Keyword(s):  
Critical Analysis ◽  
Historical Development ◽  
The Subject ◽  
The Law

In view of the interest which the subject is at present arousing, a critical analysis of the historical development of the law of insanity in its relation to divorce may be neither inopportune nor uninstructive.

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