THE COMPLEXITY OF WORD CIRCUITS

2010 ◽  
Vol 02 (04) ◽  
pp. 483-492
Author(s):  
XUE CHEN ◽  
GUANGDA HU ◽  
XIAOMING SUN
Keyword(s):  
Directed Acyclic Graph ◽  
Order Term ◽  
General Setting ◽  
Natural Generalization ◽  
Input Word ◽  
Boolean Circuit ◽  
Binary Function ◽  
Open Problems ◽  
Necessary And Sufficient ◽  
Word Function

A word circuit [1] is a directed acyclic graph in which each edge holds a w-bit word (i.e., some x ∈ {0, 1}w) and each node is a gate computing some binary function g : {0, 1}w × {0, 1}w → {0, 1}w. The following problem was studied in [1]: How many binary gates are needed to compute a ternary function f : ({0, 1}w)3 → {0, 1}w. They proved that (2 + o(1))2w binary gates are enough for any ternary function, and there exists a ternary function which requires word circuits of size (1 - o(1))2w. One of the open problems in [1] is to get these bounds tight within a low order term. In this paper we solved this problem by constructing new word circuits for ternary functions of size (1 + o(1))2w. We investigate the problem in a general setting: How many k-input word gates are needed for computing an n-input word function f : ({0, 1}w)n → {0, 1}w (here n ≥ k). We show that for any fixed n, (1 - o(1))2(n - k)w basic gates are necessary and (1 + o(1))2(n - k)w gates are sufficient (assume w is sufficiently large). Since word circuit is a natural generalization of boolean circuit, we also consider the case when w is a constant and the number of inputs n is sufficiently large. We show that [Formula: see text] basic gates are necessary and sufficient in this case.

Mixing for stationary processes with finite-order multiple Wiener-Itô integral representation

1996 ◽  
Vol 16 (5) ◽  
pp. 1087-1100
Author(s):  
Eric Slud ◽  
Daniel Chambers
Keyword(s):  
Large Class ◽  
Order Term ◽  
Sufficient Conditions ◽  
Polynomial Form ◽  
Itô Integral ◽  
Ito Integral ◽  
Weakly Mixing ◽  
Subordinated Processes ◽  
Infinite Sums ◽  
Necessary And Sufficient

abstractNecessary and sufficient analytical conditions are given for homogeneous multiple Wiener-Itô integral processes (MWIs) to be mixing, and sufficient conditions are given for mixing of general square-integrable Gaussian-subordinated processes. It is shown that every finite or infinite sum Y of MWIs (i.e. every real square-integrable stationary polynomial form in the variables of an underlying weakly mixing Gaussian process) is mixing if the process defined separately by each homogeneous-order term is mixing, and that this condition is necessary for a large class of Gaussian-subordinated processes. Moreover, for homogeneous MWIs Y1, for sums of MWIs of order ≤ 3, and for a large class of square-integrable infinite sums Y1, of MWIs, mixing holds if and only if Y2 has correlation-function decaying to zero for large lags. Several examples of the criteria for mixing are given, including a second-order homogeneous MWI, i.e. a degree two polynomial form, orthogonal to all linear forms, which has auto-correlations tending to zero for large lags but is not mixing.

scholarly journals Stability and memory-loss go hand-in-hand: three results in dynamics and computation

2020 ◽  
Vol 476 (2242) ◽  
pp. 20200563
Author(s):  
G. Manjunath
Keyword(s):  
State Of The Art ◽  
Memory Loss ◽  
General Setting ◽  
Biologically Inspired ◽  
Driven Systems ◽  
Hardware Implementations ◽  
Internal States ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Dedicated Hardware

The search for universal laws that help establish a relationship between dynamics and computation is driven by recent expansionist initiatives in biologically inspired computing. A general setting to understand both such dynamics and computation is a driven dynamical system that responds to a temporal input. Surprisingly, we find memory-loss a feature of driven systems to forget their internal states helps provide unambiguous answers to the following fundamental stability questions that have been unanswered for decades: what is necessary and sufficient so that slightly different inputs still lead to mostly similar responses? How does changing the driven system’s parameters affect stability? What is the mathematical definition of the edge-of-criticality? We anticipate our results to be timely in understanding and designing biologically inspired computers that are entering an era of dedicated hardware implementations for neuromorphic computing and state-of-the-art reservoir computing applications.

scholarly journals Hemi-slant ξ⊥-Riemannian submersions in contact geometry

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3747-3758
Author(s):  
Ramazan Sari ◽  
Mehmet Akyol
Keyword(s):  
Riemannian Manifolds ◽  
Natural Generalization ◽  
Product Manifold ◽  
Necessary And Sufficient Condition ◽  
Sasakian Manifolds ◽  
Riemannian Submersions ◽  
Space Forms ◽  
Base Manifold ◽  
Sasakian Space Forms ◽  
Necessary And Sufficient

M. A. Akyol and R. Sar? [On semi-slant ??-Riemannian submersions, Mediterr. J. Math. 14(6) (2017) 234.] defined semi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. As a generalization of the above notion and natural generalization of anti-invariant ??-Riemannian submersions, semi-invariant ??-Riemannian submersions and slant submersions, we study hemi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We obtain the geometry of foliations, give some examples and find necessary and sufficient condition for the base manifold to be a locally product manifold. Moreover, we obtain some curvature relations from Sasakian space forms between the total space, the base space and the fibres.

scholarly journals Lower Bound of Sectional Curvature of Manifold of Beta Distributions and Complete Monotonicity of Functions Involving Polygamma Functions

2020 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Lower Bound ◽  
Sectional Curvature ◽  
Sufficient Conditions ◽  
Laplace Transforms ◽  
Complete Monotonicity ◽  
Convolution Theorem ◽  
Open Problems ◽  
Polygamma Functions ◽  
Beta Distributions ◽  
Necessary And Sufficient

In the paper, by convolution theorem for the Laplace transforms and analytic techniques, the author finds necessary and sufficient conditions for complete monotonicity, monotonicity, and inequalities of several functions involving polygamma functions. By these results, the author derives a lower bound of a function related to the sectional curvature of the manifold of the beta distributions. Finally, the author poses several guesses and open problems related to monotonicity, complete monotonicity, and inequalities of several functions involving polygamma functions.

On the structure of nonstabilizer Cliford codes

2004 ◽  
Vol 4 (2) ◽  
pp. 152-160
Author(s):  
A.A. Klappenecker ◽  
M. Rotteler
Keyword(s):  
Error Control ◽  
Natural Generalization ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Error Control Codes ◽  
Stabilizer Codes ◽  
Stabilizer Code ◽  
Necessary And Sufficient ◽  
Quantum Error

Clifford codes are a class of quantum error control codes that form a natural generalization of stabilizer codes. These codes were introduced in 1996 by Knill, but only a single Clifford code was known, which was not already a stabilizer code. We derive a necessary and sufficient condition that allows one to decide when a Clifford code is a stabilizer code, and compile a table of all true Clifford codes for error groups of small order.

scholarly journals Fixed point theorems for non-self maps I

1994 ◽  
Vol 17 (4) ◽  
pp. 713-716 ◽  
Author(s):  
Troy L. Hicks ◽  
Linda Marie Saliga
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Closed Subset ◽  
Fixed Point Theorems ◽  
Complete Metric Space ◽  
Sufficient Conditions ◽  
General Setting ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Supposef:C→XwhereCis a closed subset ofX. Necessary and sufficient conditions are given forfto have a fixed point. All results hold whenXis complete metric space. Several results hold in a much more general setting.

scholarly journals Balancing Spreads of Influence in a Social Network

2020 ◽  
Vol 34 (01) ◽  
pp. 3-10 ◽  
Author(s):  
Ruben Becker ◽  
Federico Corò ◽  
Gianlorenzo D'Angelo ◽  
Hugo Gilbert
Keyword(s):  
Social Media ◽  
Optimization Problem ◽  
General Setting ◽  
Natural Generalization ◽  
Constant Factor ◽  
Cascade Model ◽  
Expected Number ◽  
Exponential Time ◽  
Independent Cascade Model ◽  
Independent Cascade

The personalization of our news consumption on social media has a tendency to reinforce our pre-existing beliefs instead of balancing our opinions. To tackle this issue, Garimella et al. (NIPS'17) modeled the spread of these viewpoints, also called campaigns, using the independent cascade model introduced by Kempe, Kleinberg and Tardos (KDD'03) and studied an optimization problem that aims to balance information exposure when two opposing campaigns propagate in a network. This paper investigates a natural generalization of this optimization problem in which μ different campaigns propagate in the network and we aim to maximize the expected number of nodes that are reached by at least ν or none of the campaigns, where μ ≥ ν ≥ 2. Following Garimella et al., despite this general setting, we also investigate a simplified one, in which campaigns propagate in a correlated manner. While for the simplified setting, we show that the problem can be approximated within a constant factor for any constant μ and ν, for the general setting, we give reductions leading to several approximation hardness results when ν ≥ 3. For instance, assuming the gap exponential time hypothesis to hold, we obtain that the problem cannot be approximated within a factor of n−g(n) for any g(n) = o(1) where n is the number of nodes in the network. We complement our hardness results with an Ω(n−1/2)-approximation algorithm for the general setting when ν = 3 and μ is arbitrary.

scholarly journals REPRESENTATION OF I(1) AND I(2) AUTOREGRESSIVE HILBERTIAN PROCESSES

2019 ◽  
Vol 36 (5) ◽  
pp. 773-802 ◽  
Author(s):  
Brendan K. Beare ◽  
Won-Ki Seo
Keyword(s):  
Separable Hilbert Space ◽  
Sufficient Conditions ◽  
General Setting ◽  
Operator Pencil ◽  
Second Order ◽  
Vector Autoregressive ◽  
Functional Time Series ◽  
Vector Autoregressive Processes ◽  
Necessary And Sufficient ◽  
Autoregressive Hilbertian Processes

We develop versions of the Granger–Johansen representation theorems for I(1) and I(2) vector autoregressive processes that apply to processes taking values in an arbitrary complex separable Hilbert space. This more general setting is of central relevance for statistical applications involving functional time series. An I(1) or I(2) solution to an autoregressive law of motion is obtained when the inverse of the autoregressive operator pencil has a pole of first or second order at one. We obtain a range of necessary and sufficient conditions for such a pole to be of first or second order. Cointegrating and attractor subspaces are characterized in terms of the behavior of the autoregressive operator pencil in a neighborhood of one.

scholarly journals Enveloping semigroups and quasi-discrete spectrum

2015 ◽  
Vol 36 (8) ◽  
pp. 2627-2660 ◽  
Author(s):  
JUHO RAUTIO
Keyword(s):  
Discrete Spectrum ◽  
American Mathematical Society ◽  
Natural Generalization ◽  
Minimal System ◽  
Topological Groups ◽  
Open Problems ◽  
Universal System ◽  
Infinite Dimensional ◽  
Enveloping Semigroups ◽  
The One

The structures of the enveloping semigroups of certain elementary finite- and infinite-dimensional distal dynamical systems are given, answering open problems posed in 1982 by Namioka [Ellis groups and compact right topological groups. Conference in Modern Analysis and Probability (New Haven, CT, 1982) (Contemporary Mathematics, 26). American Mathematical Society, Providence, RI, 1984, 295–300]. The universal minimal system with (topological) quasi-discrete spectrum is obtained from the infinite-dimensional case. It is proved that, on the one hand, a minimal system is a factor of this universal system if and only if its enveloping semigroup has quasi-discrete spectrum and that, on the other hand, such a factor need not have quasi-discrete spectrum in itself. This leads to a natural generalization of the property of having quasi-discrete spectrum, which is named the ${\mathcal{W}}$-property.

scholarly journals Some results on the asymptotic behavior of nonoscillatory solutions of differential equations with deviating arguments

1982 ◽  
Vol 32 (3) ◽  
pp. 295-317 ◽  
Author(s):  
Ch. G. Philos ◽  
Y. G. Sficas ◽  
V. A. Staikos
Keyword(s):  
Differential Equations ◽  
Asymptotic Properties ◽  
General Setting ◽  
Nonoscillatory Solution ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Nonoscillatory Solutions ◽  
Deviating Arguments ◽  
Image Position ◽  
Necessary And Sufficient

AbstractThis paper deals with some asymptotic properties of nonoscillatory solutions of a class of n-th order (n < 1) differential equations with deviationg arguments involving the so called n-th order r-derivative of the unknown function x defined bywhere ri (i = 0,1…n) are positive continous functions on [t0, ∞). The fundamental purpose of this paper is to find for any integer m, 0 < m < n – 1, a necessary and sufficient condition (depending on m) in order that three exists at least one (nonoscillatory) solution x so that the exists in R – {0} The results obtained extend some recent ones due to Philos (1978a) and they prove, in a general setting, the validity of a conjecture made by Kusano and Onose (1975).

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