We explore ideas for speeding up signature schemes that use multivariate polynomials. In particular, we propose a system with odd characteristic and a secret map of degree 2. Using odd characteristic instead of characteristic 2 has a profound effect, which we attempt to explain and also demonstrate through experiment. We discuss known attacks which could possibly topple such systems, especially algebraic attacks. After testing the resilience of these schemes against the F4 algorithm, we suggest parameters that yield acceptable security levels.
All necessary definitions will be explained and the talk should be approachable to first year graduate students.
When we need to find the function that minimizes or maximizes a convex functional, the derivative plays an important role. So when there is no meaning to the common derivative, it becomes difficult to think about solving this problem. The second moment is the one of continuity and integrability. When the functions are not Lebesgue integrable and even not Bochner integrable, the problem becomes more abstract:
How can we define and integral, the topology and the derivative so that the problem makes sense? And how will the maximum principle look like?
There are many encryption schemes that utilize the isomorphism of degree-n extensions of a finite field k and the vector space k^n. Recently, a very promising digital signature scheme was broken via matrices similar to those which represent multiplications in the extension field. We will discuss the basics and give an overview of the famous attack, and explore the apparent limitations of this attack strategy.
There has been an upsurge of interest in analyzing multivariate
time series over the past decades, because many time series arising in practice
are best considered as components of vector-valued (multivariate) time series.
Everyday, tremendous amount of nancial, biological, medical, social network
data are collected from various resources. Commonly, these data are of very
high dimensionality and moderate sample size, where traditional statistical
analysis methods perform poorly. In order to address this problem, we pro-
pose in this paper to regularize stationary time series by penalty. This will
generate sparse estimators automatically if the model is truly sparse. Consis-
tency theorems are developed for the general cases so that application from
diverse elds can bene t from our method. As an illustrating example, we
have applied our new method to a popular nancial model: the full-factor
multivariate GARCH model. Both simulation and real data results support
our theoretical deduction and will be reported in the presentation.
This is the joint work with Xiaodong Lin (University of Cincinnati).
In the past few years a new family of methods for the numerical solution of partial differential equations has become of interest, first for the engineering community and later for the mathematical community. These methods are called Meshfree or Meshless. As their name indicates, they were stimulated for the difficulties related to mesh generation which is a delicate and expensive task in many situations, like problems with complicated geometries, problems involving moving discontinuities, evolving material interfaces, multiple-scale phenomena, large material distortion and structural deformation. In all these cases and many other situations, the applicability of mesh-based methods like Finite Elements or Finite Differences is limited or even impossible. But meshfree methods have been applied with great success. In addition, the need for flexibility in the selection of approximating functions (e.g., the flexibility to use non-polynomial approximating functions), has played a significant role in the development of these methods.
In this talk I will introduce meshless methods, in particular the Element Free Galerkin (EFG) method for solving linear elliptic equations using variational principles. I will show some of the mathematical theory, briefly address implementational aspects and present some basic numerical examples.
We will talk about general optimal control problems, starting with variational problems in Banach spaces. I will talk about the notion of derivative in functional spaces and their use. We will also discuss the proof of the Pontriagin's Principle of Maximum in infinite dimensional spaces for problems with phase constraints
In this talk we survey recent results in the study of bilipschitz homogeneous (BLH) Jordan curves. In particular, we discuss an quantitative extension of Bishop's result, which states that an unbounded BLH Jordan curve is of bounded turning. We also consider the relationship between BLH curves and Mobius maps. Lastly, we extend results of Rohde, providing a catalogue of all unbounded BLH curves, up to certain bilipschitz maps.
In this talk, I will present the method of exchangeable pairs introduced by Stein in 1972 to get the bounds on the distance between the Binomial distribution and its approximating, Poisson distribution.
In this talk I will discuss the duality between the $L^p$ regularity and the $L^{p\prime}$ Dirichlet problem for second order elliptic systems with constant coefficients on bounded Lipschitz domains. In particular I will show that the solvability of the $L^p$ regularity problems is equivalent to the solvability of the $L^{p\prime}$ Dirichlet problem. This is a joint work with Dr. Zhongwei Shen.
In this talk I will show a historical point of view of the origin of Boussinesq Equation (a type of wave equation), the relationship with the KdV equation and explain the well-posedness problem for the Boussinesq equation and how it has been studied since its origin.
I will describe induction-on-scales proofs of some basic (and some perhaps less so) inequalities for several well-known functional classes. The key to this method of proof is to establish the existence, size and convexity of a certain function (often termed "Bellman function"). To arrive at the inequality, the function is "unwrapped" along a binary tree rooted in the measure space on which the functional class in question is defined.
In this talk, I will give a general, easy accessible and hopefully interesting introduction to the regularization techniques for multivariate stationary time series. Multivariate time series models are very popular tools in many aspects of scientific research: finance, econometrics, biotechnology, meteorology, social networks, etc, etc. The theme of my research is to solve the 'curse of dimensionality' of such models so that we get much relieved of the heavy computational burden yet still able to benefit from the intrinsically complex structure. This time I will mainly focus on the origin of the problem and the methodology history in the literature, so even for audience totally outside the field of statistics or time series it should be perfectly understandable. I will also provide with simulation and real data results to 'show off' the advantages of our method, but that part should be even more understandable. I can also talk briefly about the technical proofs, depending on the requests from the audience.
We define amenability for discrete groups and we see a few examples and some connections to geometric group theory. The last part of the talk will be a discussion of a specific amenable group, the lamplighter group.
Bayesian network, also called Graphical Model, is a bridge between graph theory and statistics. A Bayesian network is a directed acyclic graph that encodes causal relationships among a set of random variables by the existence of edges in the graph. Starting with Bayes' Rule, this talk is an brief introduction to Bayesian network modeling, including the definition and factorization theorem of Bayesian network, and three main topic for Bayesian networks: parameter estimation, inference, and structure learning. A simple bank fraud example will be used throughout the talk. I will bring a power point of about 20 slides and an easy sample data to illustrate why Bayesian network is interesting.
We will discuss some ideas behind a successful job search and how to avoid some common pitfalls along the way. In addition to research and teaching statements, we will discuss how technology has impacted the job search.
I will talk about my own experience in the math department for the past 4 years, pointing out at research opportunities. In particular I will talk about funding, travel and summer opportunities. I invite other senior students to join us and share their own experience.
One of the most developing areas of contemporary analysis is geometric function theory. Its main goals include finding counterparts of harmonic functions and mappings in the nonlinear setting and employ these generalizations in applications, e.g. in nonlinear elasticity theory, non-newtonian fluid dynamics, image processing, etc.
The purpose of the talk is to introduce the fundamental object of nonlinear potential theory, the so-called p-harmonic operator and related p-harmonic equation. If p=2 we retrieve the harmonic case, but in general the geometry of p-harmonic world is much more complicated than the harmonic one. We explain the basic properties of the nonlinear Laplacian and show the unexpected and fruitful interplay between PDEs and the class of mappings of finite distortion. We also introduce p-harmonic maps and equations with nonstandard growth conditions - another current areas of activity in nonlinear analysis.
In this talk we characterize fractal chordarc curves in Euclidean space by the fact that they remain bilipschitz homogeneous under inversion. The relevant background and definitions will be discussed, and it should be quite accessible.
In this talk, I will present the method of exchangeable pairs introduced by Stein in 1972 to get the bounds on the distance between the Binomial distribution and its approximating, Poisson distribution.
This presentation is for a general audience. I will introduce the definition of length and weighted length metrics in general setting metric spaces. Then, I will discuss the so-called quasihyperbolic metric as an example of a weighted length metric in R^n. Finally, I will introduce the Ferrand metric which is the metric I have been studying. The Ferrand metric is Mobius invariant and comparable to quasihyperbolic metric. Also, I will show some geometric properties of quasihyperbolic metric depending on the time.