An Elementary Proof of Suslin Reciprocity

This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case of number fields K, it is a theorem for function fields, due to Drinfeld and L. Lafforgue. The key innovation in this case is Drinfeld's notion of an X-shtuka (or simply shtuka). The Langlands correspondence for X is obtained by studying moduli spaces of shtukas. A large part of this course is about the definition of perfectoid spaces and diamonds. There is an important special case where the moduli spaces of shtukas are classical rigid-analytic spaces. This is the case of local Shimura varieties. Some examples of these are the Rapoport-Zink spaces.

ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

Stochastics and Dynamics ◽

10.1142/s0219493704000900 ◽

2004 ◽

Vol 04 (01) ◽

pp. 63-76 ◽

Cited By ~ 24

Author(s):

OLIVER JENKINSON

Keyword(s):

Hausdorff Dimension ◽

Finite Subset ◽

Continued Fraction ◽

Cantor Set ◽

Computational Method ◽

Accurate Approximation ◽

Natural Numbers ◽

Important Special Case ◽

The One ◽

Special Case

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

On the Hyperplane Sections Through two Given Points of an Algebraic Variety

Canadian Journal of Mathematics ◽

10.4153/cjm-1970-016-x ◽

1970 ◽

Vol 22 (1) ◽

pp. 128-133 ◽

Cited By ~ 1

Author(s):

Wei-Eihn Kuan

Keyword(s):

Simple Proof ◽

Hyperplane Section ◽

Rational Points ◽

Infinite Field ◽

Important Special Case ◽

Irreducible Curve ◽

Irreducible Variety ◽

Dimension 2 ◽

Hyperplane Sections ◽

Special Case

1. Let k be an infinite field and let V/k be an irreducible variety of dimension ≧ 2 in a projective n-space Pn over k. Let P and Q be two k-rational points on V In this paper, we describe ideal-theoretically the generic hyperplane section of V through P and Q (Theorem 1) and prove that the section is almost always an absolutely irreducible variety over k1/pe if V/k is absolutely irreducible (Theorem 3). As an application (Theorem 4), we give a new simple proof of an important special case of the existence of a curve connecting two rational points of an absolutely irreducible variety [4], namely any two k-rational points on V/k can be connected by an irreducible curve.I wish to thank Professor A. Seidenberg for his continued advice and encouragement on my thesis research.

Deception through Half-Truths

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v34i06.6570 ◽

2020 ◽

Vol 34 (06) ◽

pp. 10110-10117

Author(s):

Andrew Estornell ◽

Sanmay Das ◽

Yevgeniy Vorobeychik

Keyword(s):

Polynomial Time ◽

Transition Function ◽

Np Hard ◽

Fundamental Issue ◽

Fake News ◽

False Information ◽

Important Special Case ◽

Bayes Network ◽

Special Case

Deception is a fundamental issue across a diverse array of settings, from cybersecurity, where decoys (e.g., honeypots) are an important tool, to politics that can feature politically motivated “leaks” and fake news about candidates. Typical considerations of deception view it as providing false information. However, just as important but less frequently studied is a more tacit form where information is strategically hidden or leaked. We consider the problem of how much an adversary can affect a principal's decision by “half-truths”, that is, by masking or hiding bits of information, when the principal is oblivious to the presence of the adversary. The principal's problem can be modeled as one of predicting future states of variables in a dynamic Bayes network, and we show that, while theoretically the principal's decisions can be made arbitrarily bad, the optimal attack is NP-hard to approximate, even under strong assumptions favoring the attacker. However, we also describe an important special case where the dependency of future states on past states is additive, in which we can efficiently compute an approximately optimal attack. Moreover, in networks with a linear transition function we can solve the problem optimally in polynomial time.

On Operator Algebras and Invariant Subspaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-129-9 ◽

1969 ◽

Vol 21 ◽

pp. 1178-1181 ◽

Cited By ~ 10

Author(s):

Chandler Davis ◽

Heydar Radjavi ◽

Peter Rosenthal

Keyword(s):

Hilbert Space ◽

Elementary Proof ◽

Operator Algebras ◽

Equivalent Form ◽

Invariant Subspaces ◽

Complex Hilbert Space ◽

Closed Algebra ◽

Weakly Closed ◽

Special Case

If is a collection of operators on the complex Hilbert space , then the lattice of all subspaces of which are invariant under every operator in is denoted by Lat . An algebra of operators on is defined (3; 4) to be reflexive if for every operator B on the inclusion Lat ⊆ Lat B implies .Arveson (1) has proved the following theorem. (The abbreviation “m.a.s.a.” stands for “maximal abelian self-adjoint algebra”.)ARVESON's THEOREM. Ifis a weakly closed algebra which contains an m.a.s.a.y and if Lat, then is the algebra of all operators on .A generalization of Arveson's Theorem was given in (3). Another generalization is Theorem 2 below, an equivalent form of which is Corollary 3. This theorem was motivated by the following very elementary proof of a special case of Arveson's Theorem.

The New qd Algorithms

Acta Numerica ◽

10.1017/s0962492900002580 ◽

1995 ◽

Vol 4 ◽

pp. 459-491 ◽

Cited By ~ 26

Author(s):

Beresford N. Parlett

Keyword(s):

Singular Values ◽

Eigenvalues And Eigenvectors ◽

Triangular Factorization ◽

Matrix Computation ◽

Important Special Case ◽

Singular Vectors ◽

Tridiagonal Matrices ◽

Recent Origin ◽

Special Case

Let us think about ways to find both eigenvalues and eigenvectors of tridiagonal matrices. An important special case is the computation of singular values and singular vectors of bidiagonal matrices. The discussion is addressed both to specialists in matrix computation and to other scientists whose main interests lie elsewhere. The reason for hoping to communicate with two such diverse sets of readers at the same time is that the content of the survey, though of recent origin, is quite elementary and does not demand familiarity with much beyond triangular factorization and the Gram-Schmidt process for orthogonalizing a set of vectors. For some readers the survey will cover familiar territory but from a novel perspective. The justification for presenting these ideas is that they lead to new variations of current methods that run a lot faster while achieving greater accuracy.

On the number of excursion sets of planar Gaussian fields

Probability Theory and Related Fields ◽

10.1007/s00440-020-00984-9 ◽

2020 ◽

Vol 178 (3-4) ◽

pp. 655-698

Author(s):

Dmitry Beliaev ◽

Michael McAuley ◽

Stephen Muirhead

Keyword(s):

Plane Wave ◽

Lower Bounds ◽

Upper And Lower Bounds ◽

Gaussian Fields ◽

Important Special Case ◽

Level Densities ◽

Excursion Sets ◽

Nodal Set ◽

Different Types ◽

Special Case

Abstract The Nazarov–Sodin constant describes the average number of nodal set components of smooth Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important special case of the random plane wave.

A representation for discrete distributions by equiprobable mixtures

Journal of Applied Probability ◽

10.1017/s0021900200046842 ◽

1980 ◽

Vol 17 (01) ◽

pp. 102-111 ◽

Cited By ~ 1

Author(s):

Arthur V. Peterson ◽

Richard A. Kronmal

Keyword(s):

Random Variables ◽

Discrete Distribution ◽

Discrete Distributions ◽

Important Special Case ◽

Discrete Random Variables ◽

Special Case

We obtain a representation of an arbitrary discrete distribution with n mass points by an equiprobable mixture of r distributions, each of which has no more than a (≧2) mass points, where r is the smallest integer greater than or equal to (n – 1)/(a – 1). An application to the generation of discrete random variables on a computer is described, which has as an important special case Walker's (1977) alias method.

On the inverses of Kasami and Bracken–Leander exponents

Designs Codes and Cryptography ◽

10.1007/s10623-020-00804-0 ◽

2020 ◽

Vol 88 (12) ◽

pp. 2597-2621

Author(s):

Lukas Kölsch

Keyword(s):

Algebraic Degree ◽

Binary Representation ◽

Important Special Case ◽

Kasami Function ◽

Special Case

Abstract We explicitly determine the binary representation of the inverse of all Kasami exponents $$K_r=2^{2r}-2^r+1$$ K r = 2 2 r - 2 r + 1 modulo $$2^n-1$$ 2 n - 1 for all possible values of n and r. This includes as an important special case the APN Kasami exponents with $$\gcd (r,n)=1$$ gcd ( r , n ) = 1 . As a corollary, we determine the algebraic degree of the inverses of the Kasami functions. In particular, we show that the inverse of an APN Kasami function on $${\mathbb {F}}_{2^n}$$ F 2 n always has algebraic degree $$\frac{n+1}{2}$$ n + 1 2 if $$n\equiv 0 \pmod 3$$ n ≡ 0 ( mod 3 ) . For $$n\not \equiv 0 \pmod 3$$ n ≢ 0 ( mod 3 ) we prove that the algebraic degree is bounded from below by $$\frac{n}{3}$$ n 3 . We consider Kasami exponents whose inverses are quadratic exponents or Kasami exponents. We also determine the binary representation of the inverse of the Bracken–Leander exponent $$BL_r=2^{2r}+2^r+1$$ B L r = 2 2 r + 2 r + 1 modulo $$2^n-1$$ 2 n - 1 where $$n=4r$$ n = 4 r and r odd. We show that the algebraic degree of the inverse of the Bracken–Leander function is $$\frac{n+2}{2}$$ n + 2 2 .

Generalized Cycloidal Motion

Journal of Engineering for Industry ◽

10.1115/1.3427862 ◽

1971 ◽

Vol 93 (1) ◽

pp. 131-139 ◽

Cited By ~ 1

Author(s):

D. Spragg ◽

D. Tesar

Keyword(s):

Design Methods ◽

Important Special Case ◽

Mechanical Advantage ◽

Center Point ◽

Wheel Rolling ◽

Special Case

The motion coefficients and design methods for a wheel rolling externally or internally to a fixed wheel are given in Part I for linkage constraint satisfying up to 6 multiply separated positions. Part II provides design equations and graphical procedures in terms of the cubic circle and center point curves based on four symmetrical, multiply separated positions of the moving wheel. The important special case of an angular-circular transformer mechanism with constant mechanical advantage is treated in detail for four positions in Part III.