scholarly journals Singular tuples of matrices is not a null cone (and the symmetries of algebraic varieties)

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Visu Makam ◽  
Avi Wigderson
Keyword(s):  
Complexity Theory ◽  
Reductive Group ◽  
Deterministic Algorithm ◽  
Null Cone ◽  
Theory Of Computation ◽  
Algebraic Varieties ◽  
Singular Matrices ◽  
Special Case ◽  
Group Of Symmetries ◽  
General Method

Abstract The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory: SING n , m {{\rm SING}_{n,m}} , consisting of all m-tuples of n × n {n\times n} complex matrices which span only singular matrices. In particular, an efficient deterministic algorithm testing membership in SING n , m {{\rm SING}_{n,m}} will imply super-polynomial circuit lower bounds, a holy grail of the theory of computation. A sequence of recent works suggests such efficient algorithms for memberships in a general class of algebraic varieties, namely the null cones of linear group actions. Can this be used for the problem above? Our main result is negative: SING n , m {{\rm SING}_{n,m}} is not the null cone of any (reductive) group action! This stands in stark contrast to a non-commutative analog of this variety, and points to an inherent structural difficulty of SING n , m {{\rm SING}_{n,m}} . To prove this result, we identify precisely the group of symmetries of SING n , m {{\rm SING}_{n,m}} . We find this characterization, and the tools we introduce to prove it, of independent interest. Our work significantly generalizes a result of Frobenius for the special case m = 1 {m=1} , and suggests a general method for determining the symmetries of algebraic varieties.

Distributed Systems

2020 ◽  
pp. 143-198
Author(s):  
Andreas Bolfing
Keyword(s):  
Distributed Systems ◽  
Fault Tolerant ◽  
Deterministic Algorithm ◽  
Impossibility Result ◽  
Software Systems ◽  
Trade Off ◽  
Byzantine Failures ◽  
The Individual ◽  
Special Case

Chapter 5 considers distributed systems by their properties. The first section studies the classification of software systems, which is usually distinguished in centralized, decentralized and distributed systems. It studies the differences between these three major approaches, showing there is a rather multidimensional classification instead of a linear one. The most important case are distributed systems that enable spreading of computational tasks across several autonomous, independently acting computational entities. A very important result of this case is the CAP theorem that considers the trade-off between consistency, availability and partition tolerance. The last section deals with the possibility to reach consensus in distributed systems, discussing how fault tolerant consensus mechanisms enable mutual agreement among the individual entities in presence of failures. One very special case are so-called Byzantine failures that are discussed in great detail. The main result is the so-called FLP Impossibility Result which states that there is no deterministic algorithm that guarantees solution to the consensus problem in the asynchronous case. The chapter concludes by considering practical solutions that circumvent the impossibility result in order to reach consensus.

scholarly journals Analytic Continuation of Equivariant Distributions

2018 ◽  
Vol 2019 (23) ◽  
pp. 7160-7192 ◽  
Author(s):  
Dmitry Gourevitch ◽  
Siddhartha Sahi ◽  
Eitan Sayag
Keyword(s):  
Analytic Continuation ◽  
Reductive Group ◽  
Principal Series ◽  
Real Algebraic Varieties ◽  
Series Representations ◽  
Degenerate Principal Series ◽  
Special Case ◽  
Adic Case ◽  
Principal Series Representations ◽  
Whittaker Models

Abstract We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein’s theory of analytic continuation of holonomic distributions. We use this to construct H-equivariant functionals on principal series representations of G, where G is a real reductive group and H is an algebraic subgroup. We also deduce the existence of generalized Whittaker models for degenerate principal series representations. As a special case, this gives short proofs of existence of Whittaker models on principal series representations and of analytic continuation of standard intertwining operators. Finally, we extend our constructions to the p-adic case using a recent result of Hong and Sun.

scholarly journals Budget Feasible Mechanisms on Matroids

Algorithmica ◽  
2020 ◽  
Author(s):  
Stefano Leonardi ◽  
Gianpiero Monaco ◽  
Piotr Sankowski ◽  
Qiang Zhang
Keyword(s):  
Polynomial Time ◽  
Private Information ◽  
Independent Set ◽  
Deterministic Algorithm ◽  
Matroid Intersection ◽  
Practical Applications ◽  
Incentive Compatible ◽  
The Cost ◽  
Special Case ◽  
Budget Feasible

AbstractMotivated by many practical applications, in this paper we study budget feasible mechanisms with the goal of procuring an independent set of a matroid. More specifically, we are given a matroid $${\mathcal {M}}=(E,{\mathcal {I}})$$ M = ( E , I ) . Each element of the ground set E is controlled by a selfish agent and the cost of the element is private information of the agent itself. A budget limited buyer has additive valuations over the elements of E. The goal is to design an incentive compatible budget feasible mechanism which procures an independent set of the matroid of largest possible value. We also consider the more general case of the pair $${\mathcal {M}}=(E,{\mathcal {I}})$$ M = ( E , I ) satisfying only the hereditary property. This includes matroids as well as matroid intersection. We show that, given a polynomial time deterministic algorithm that returns an $$\alpha $$ α -approximation to the problem of finding a maximum-value independent set in $${\mathcal {M}}$$ M , there exists an individually rational, truthful and budget feasible mechanism which is $$(3\alpha +1)$$ ( 3 α + 1 ) -approximated and runs in polynomial time, thus yielding also a 4-approximation for the special case of matroids.

scholarly journals Products of Variables in Structural Equation Models: Multiplicative Reticular Action Models

2019 ◽  
Author(s):  
Steven M. Boker ◽  
Timo von Oertzen ◽  
Andreas Markus Brandmaier
Keyword(s):  
Latent Variables ◽  
Latent Variable ◽  
Structural Equation ◽  
Structural Equation Models ◽  
Equation Model ◽  
Random Coefficients ◽  
Action Model ◽  
Action Models ◽  
Special Case ◽  
General Method

A general method is introduced in which variables that are products of other variables in the context of a structural equation model (SEM) can be decomposed into the sources of variance due to the multiplicands. The result is a new category of SEM which we call a Multiplicative Reticular Action Model (XRAM). XRAM can include interactions between latent variables, multilevel random coefficients, latent variable moderators, and novel constructs such as factors of paths and twin genetic decomposition of multilevel random coefficients. The method relies on an assumption that all variance sources in a model can be decomposed into linear combinations of independent normal standardized variables. Although the distribution of a variable that is an outcome of multiplication between other variables is not normal, the assumption is that it can be decomposed into sources that are normal if one takes into account the non-normality induced by the multiplication. The method is applied to an example to show how in a special case it is equivalent to known unbiased and efficient estimators in the statistical literature. Two simulations are presented that demonstrate the precision of the approximation and implement the method to estimate parameters in a multilevel autoregressive framework.

scholarly journals IV. Mathematical contributions to the theory of evolution.—V. On the reconstruction of the stature of prehistoric races

1899 ◽  
Vol 192 ◽  
pp. 169-244 ◽  
Keyword(s):  
Lower Limbs ◽  
Long Bones ◽  
Upper Limbs ◽  
Theory Of Evolution ◽  
Remarkable Degree ◽  
The Mean ◽  
Standard Deviations ◽  
Special Case ◽  
General Method ◽  
Local Race

(1.) The object of this paper is to show, by the use of a special case as illustration, the true limits within which it is possible to reconstruct the parts of an extinct race from a knowledge of the size of a few organs or bones, when complete measurements have been or can be made for an allied and still extant race. The illustration I have taken is one of considerable interest in itself, and has been considered from a variety of standpoints by a long series of investigators. But I wish it to be considered purely as an illustration of a general method. What is here done for stature from long bones is equally applicable to other organs in Man. We might reconstruct in the same manner the dimensions of the hand from a knowledge of any of the finger bones, or the bones of the upper limbs from a knowledge of the bones of the lower limbs. Further, we need not confine our attention to Man, but can predict, with what often amounts to a remarkable degree of accuracy, the dimensions of the organs of one local race of any species from a knowledge of a considerable number of organs in a second local race, and of only one or two organs of the first. The importance of this result for the reconstruction of fossil or prehistoric races will be obvious. What we need for any such reconstruction are the following data:— ( a .) The mean sizes, the variabilities (standard-deviations), and the correlations of as many organs in an extant allied race as it is possible conveniently to measure. When the correlations of the organs under consideration are high ( e. g. , the long bones in Man), fifty to a hundred individuals may be sufficient; in other cases it is desirable that several hundred at least should be measured.

Bases for the Prime Ideals Associated with Certain Classes of Algebraic Varieties

1943 ◽  
Vol 39 (1) ◽  
pp. 35-48 ◽  
Author(s):  
L. S. Goddard
Keyword(s):  
Prime Ideal ◽  
Finite Basis ◽  
Ideal Theory ◽  
Prime Ideals ◽  
Algebraic Varieties ◽  
Twisted Cubic ◽  
Cubic Curve ◽  
Principal Theorem ◽  
General Method ◽  
The Ideal

The fact that the prime ideal associated with a given irreducible algebraic variety has a finite basis is a pure existence theorem. Only in a few isolated particular cases has the base for the ideal been found, and there appears to be no general method for determining the base which can be carried out in practice. Hilbert, who initiated the theory, proved that the prime ideal defining the ordinary twisted cubic curve has a base consisting of three quadrics, and contributions to the ideal theory of algebraic varieties have been made by König, Lasker, Macaulay and, more recently, by Zariski. A good summary, from the viewpoint of a geometer, is given by Bertini [(1), Chapter XII]. However, the tendency has been towards the development of the pure theory. In the following paper we actually find the bases for the prime ideals associated with certain classes of algebraic varieties. The paper falls into two parts. In Part I there is proved a theorem (the Principal Theorem) of wide generality, and then examples are given of some classes of varieties satisfying the conditions of the theorem. In Part II we find the base for the prime ideals associated with Veronesean varieties and varieties of Segre. The latter are particularly interesting since they represent (1, 1), without exception, the points of a multiply-projective space.

Cyclic ether synthesis: sulfenyletherification with benzenesulfenyl chloride/N, N-diisopropylethylamine and sulfenate ester cycloadditions

1987 ◽  
Vol 65 (8) ◽  
pp. 1833-1837 ◽  
Author(s):  
Sudersan M. Tuladhar ◽  
Alex G. Fallis
Keyword(s):  
Cyclic Ether ◽  
Cyclic Ethers ◽  
Nucleophilic Displacement ◽  
Ether Synthesis ◽  
Special Case ◽  
General Method ◽  
Episulfonium Ion

A general method for the formation of the cyclic ethers 9, 10, 13, 14, 16, and 28 and the lactone 30 is described. The procedure employs benzenesulfenyl chloride prepared insitu in acetonitrile and N,N-diisopropylethylamine to generate a thiiranium (episulfonium) ion intermediate from which the cyclic products arise by internal nucleophilic displacement. In the special case of the norbornene alcohols 1 and 2 the oxetanes 5 and 6 are formed by intramolecular sulfenate ester cycloaddition.

Spinwellen in dünnen ferromagnetischen Schichten

1964 ◽  
Vol 19 (13) ◽  
pp. 1567-1580 ◽  
Author(s):  
Rainer Jelitto
Keyword(s):  
Difference Equation ◽  
Surface States ◽  
Subsequent Paper ◽  
Cubic Lattice ◽  
Starting Point ◽  
Transcendental Equations ◽  
The One ◽  
The Given ◽  
Special Case ◽  
General Method

This paper is concerned with an ideal spin-l/2-HEisENBERG-model for thin ferromagnetic films. A general method is given for the calculation of the one-spinwave eigenstates and their spectrum in dependence on the lattice type and the orientation of the surfaces of the film. The function that characterises the shape of the spinwave perpendicular to the film must fulfil a linear eigenvalue-difference-equation as well as a set of boundary conditions.For next-neighbour interactions this system may be evaluated for an especially simple case. For it spinwavestates of the form of cos-sin-functions as well as surface states are found. Their momenta are given by some transcendental equations, which are discussed.For all other cases the given difference-equation cannot be solved in a closed form, but at any rate it is a starting point for numerical calculations.In a subsequent paper it will be shown that the special case mentioned above covers some important surface orientations of the cubic lattice types. For films of these orientations the dependence of the magnetization on temperature and thickness of the film will be derived from the spinwave spectra.

scholarly journals Strong Approximation and a Conjecture of Harpaz and Wittenberg

2017 ◽  
Vol 2019 (14) ◽  
pp. 4340-4369
Author(s):  
Tim D Browning ◽  
Damaris Schindler
Keyword(s):  
Strong Approximation ◽  
Algebraic Varieties ◽  
Special Case

Abstract We study strong approximation for some algebraic varieties over $\mathbb{Q}$ which are defined using norm forms. This allows us to confirm a special case of a conjecture due to Harpaz and Wittenberg.

Cyclic homology and path algebra resolutions

1989 ◽  
Vol 106 (1) ◽  
pp. 57-66 ◽  
Author(s):  
Dave Benson
Keyword(s):  
Characteristic Zero ◽  
Directed Graphs ◽  
Path Algebra ◽  
Cyclic Homology ◽  
Free Algebras ◽  
Path Algebras ◽  
Special Case ◽  
General Method

It follows from a theorem of Loday and Quillen (proposition 5·4 of [6]) that one may calculate the cyclic homology of an algebra in characteristic zero by taking a semisimplicial resolution by free algebras, quotienting out commutators and then taking homology of the resulting complex. In this paper we explain how this is a special case of a more general method based on resolutions by path algebras of directed graphs. The Loday–Quillen result may be seen as the case where the graph has only one vertex.

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