Mathematik
Refine
Year of publication
- 2008 (10)
- 2011 (8)
- 2022 (7)
- 2023 (6)
- 2002 (5)
- 2020 (5)
- 2021 (5)
- 1994 (4)
- 2001 (4)
- 2017 (4)
- 1921 (3)
- 1999 (3)
- 2000 (3)
- 2003 (3)
- 2004 (3)
- 2010 (3)
- 2015 (3)
- 2019 (3)
- 1996 (2)
- 1998 (2)
- 2005 (2)
- 2006 (2)
- 2012 (2)
- 2024 (2)
- 1897 (1)
- 1903 (1)
- 1906 (1)
- 1909 (1)
- 1912 (1)
- 1918 (1)
- 1919 (1)
- 1922 (1)
- 1924 (1)
- 1934 (1)
- 1978 (1)
- 1991 (1)
- 1992 (1)
- 1997 (1)
- 2013 (1)
- 2014 (1)
- 2016 (1)
- 2018 (1)
Document Type
- Article (112) (remove)
Has Fulltext
- yes (112)
Is part of the Bibliography
- no (112)
Keywords
- Biographie (2)
- Brownian motion (2)
- Finanzmathematik (2)
- Frankfurt <Main> / Universität (2)
- Krein space (2)
- Mathematik (2)
- Mathematiker (2)
- Perception (2)
- Statistik (2)
- Stochastik (2)
Institute
- Mathematik (112)
- Informatik (13)
- Medizin (2)
- Frankfurt Institute for Advanced Studies (FIAS) (1)
- MPI für Hirnforschung (1)
- MPI für empirische Ästhetik (1)
- Physik (1)
- Präsidium (1)
- Psychologie (1)
The existence of a mean-square continuous strong solution is established for vector-valued Itö stochastic differential equations with a discontinuous drift coefficient, which is an increasing function, and with a Lipschitz continuous diffusion coefficient. A scalar stochastic differential equation with the Heaviside function as its drift coefficient is considered as an example. Upper and lower solutions are used in the proof.
Considered are the classes QL (quasilinear) and NQL (nondet quasllmear) of all those problems that can be solved by deterministic (nondetermlnlsttc, respectively) Turmg machines in time O(n(log n) ~) for some k Effloent algorithms have time bounds of th~s type, it is argued. Many of the "exhausUve search" type problems such as satlsflablhty and colorabdlty are complete in NQL with respect to reductions that take O(n(log n) k) steps This lmphes that QL = NQL iff satisfiabdlty is m QL CR CATEGORIES: 5.25
A memory checker for a data structure provides a method to check that the output of the data structure operations is consistent with the input even if the data is stored on some insecure medium. In [8] we present a general solution for all data structures that are based on insert(i,v) and delete(j) commands. In particular this includes stacks, queues, deques (double-ended queues) and lists. Here, we describe more time and space efficient solutions for stacks, queues and deques. Each algorithm takes only a single function evaluation of a pseudorandomlike function like DES or a collision-free hash function like MD5 or SHA for each push/pop resp. enqueue/dequeue command making our methods applicable to smart cards.
We analyse a continued fraction algorithm (abbreviated CFA) for arbitrary dimension n showing that it produces simultaneous diophantine approximations which are up to the factor 2^((n+2)/4) best possible. Given a real vector x=(x_1,...,x_{n-1},1) in R^n this CFA generates a sequence of vectors (p_1^(k),...,p_{n-1}^(k),q^(k)) in Z^n, k=1,2,... with increasing integers |q^{(k)}| satisfying for i=1,...,n-1 | x_i - p_i^(k)/q^(k) | <= 2^((n+2)/4) sqrt(1+x_i^2) |q^(k)|^(1+1/(n-1)) By a theorem of Dirichlet this bound is best possible in that the exponent 1+1/(n-1) can in general not be increased.
Parallel FFT-hashing
(1994)
We propose two families of scalable hash functions for collision resistant hashing that are highly parallel and based on the generalized fast Fourier transform (FFT). FFT hashing is based on multipermutations. This is a basic cryptographic primitive for perfect generation of diffusion and confusion which generalizes the boxes of the classic FFT. The slower FFT hash functions iterate a compression function. For the faster FFT hash functions all rounds are alike with the same number of message words entering each round.
Let b1, . . . , bm 2 IRn be an arbitrary basis of lattice L that is a block Korkin Zolotarev basis with block size ¯ and let ¸i(L) denote the successive minima of lattice L. We prove that for i = 1, . . . ,m 4 i + 3 ° 2 i 1 ¯ 1 ¯ · kbik2/¸i(L)2 · ° 2m i ¯ 1 ¯ i + 3 4 where °¯ is the Hermite constant. For ¯ = 3 we establish the optimal upper bound kb1k2/¸1(L)2 · µ3 2¶m 1 2 1 and we present block Korkin Zolotarev lattice bases for which this bound is tight. We improve the Nearest Plane Algorithm of Babai (1986) using block Korkin Zolotarev bases. Given a block Korkin Zolotarev basis b1, . . . , bm with block size ¯ and x 2 L(b1, . . . , bm) a lattice point v can be found in time ¯O(¯) satisfying kx vk2 · m° 2m ¯ 1 ¯ minu2L kx uk2.
Let G be a group of prime order q with generator g. We study hardcore subsets H is include in G of the discrete logarithm (DL) log g in the model of generic algorithms. In this model we count group operations such as multiplication, division while computations with non-group data are for free. It is known from Nechaev (1994) and Shoup (1997) that generic DL-algorithms for the entire group G must perform p2q generic steps. We show that DL-algorithms for small subsets H is include in G require m/ 2 + o(m) generic steps for almost all H of size #H = m with m <= sqrt(q). Conversely, m/2 + 1 generic steps are su±cient for all H is include in G of even size m. Our main result justifies to generate secret DL-keys from seeds that are only 1/2 * log2 q bits long.
We show that P(n)*(P(n)) for p = 2 with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation Epsilon nor the coproduct Delta are multiplicative. As a consequence the algebra structure of P(n)*(P(n)) is slightly different from what was supposed to be the case. We give formulas for Epsilon(xy) and Delta(xy) and show that the inversion of the formal group of P(n) is induced by an antimultiplicative involution Xi : P(n) -> P(n). Some consequences for multiplicative and antimultiplicative automorphisms of K(n) for p = 2 are also discussed.
The general subset sum problem is NP-complete. However, there are two algorithms, one due to Brickell and the other to Lagarias and Odlyzko, which in polynomial time solve almost all subset sum problems of sufficiently low density. Both methods rely on basis reduction algorithms to find short nonzero vectors in special lattices. The Lagarias-Odlyzko algorithm would solve almost all subset sum problems of density < 0.6463 . . . in polynomial time if it could invoke a polynomial-time algorithm for finding the shortest non-zero vector in a lattice. This paper presents two modifications of that algorithm, either one of which would solve almost all problems of density < 0.9408 . . . if it could find shortest non-zero vectors in lattices. These modifications also yield dramatic improvements in practice when they are combined with known lattice basis reduction algorithms.
Public key signature schemes are necessary for the access control to communication networks and for proving the authenticity of sensitive messages such as electronic fund transfers. Since the invention of the RSA scheme by Rivest, Shamir and Adleman (1978) research has focused on improving the e±ciency of these schemes. In this paper we present an efficient algorithm for generating public key signatures which is particularly suited for interactions between smart cards and terminals.
We present a novel parallel one-more signature forgery against blind Okamoto-Schnorr and blind Schnorr signatures in which an attacker interacts some times with a legitimate signer and produces from these interactions signatures. Security against the new attack requires that the following ROS-problem is intractable: find an overdetermined, solvable system of linear equations modulo with random inhomogenities (right sides). There is an inherent weakness in the security result of POINTCHEVAL AND STERN. Theorem 26 [PS00] does not cover attacks with 4 parallel interactions for elliptic curves of order 2200. That would require the intractability of the ROS-problem, a plausible but novel complexity assumption. Conversely, assuming the intractability of the ROS-problem, we show that Schnorr signatures are secure in the random oracle and generic group model against the one-more signature forgery.
We reconsider estimates for the heat kernel on weighted graphs recently found by Metzger and Stollmann. In the case that the weights satisfy a positive lower bound as well as a finite upper bound, we obtain a specialized lower estimate and a proper generalization of a previous upper estimate. Reviews: Math. Rev. 1979406, Zbl. Math. 0934.46042
We study the quantum Zeno effect in quantum statistical mechanics within the operator algebraic framework. We formulate a condition for the appearance of the effect in W*-dynamical systems, in terms of the short-time behaviour of the dynamics. Examples of quantum spin systems show that this condition can be effectively applied to quantum statistical mechanical models. Furthermore, we derive an explicit form of the Zeno generator, and use it to construct Gibbs equilibrium states for the Zeno dynamics. As a concrete example, we consider the X-Y model, for which we show that a frequent measurement at a microscopic level, e.g. a single lattice site, can produce a macroscopic effect in changing the global equilibrium. PACS - Klassifikation: 03.65.Xp, 05.30.-d, 02.30. See the corresponding papers: Schmidt, Andreas U.: "Zeno Dynamics of von Neumann Algebras" and "Mathematics of the Quantum Zeno Effect" and the talk "Zeno Dynamics in Quantum Statistical Mechanics" - http://publikationen.ub.uni-frankfurt.de/volltexte/2005/1167/
The dynamical quantum Zeno effect is studied in the context of von Neumann algebras. It is shown that the Zeno dynamics coincides with the modular dynamics of a localized subalgebra. This relates the modular operator of that subalgebra to the modular operator of the original algebra by a variant of the Kato-Lie-Trotter product formula.
We present a method for the construction of a Krein space completion for spaces of test functions, equipped with an indefinite inner product induced by a kernel which is more singular than a distribution of finite order. This generalizes a regularization method for infrared singularities in quantum field theory, introduced by G. Morchio and F. Strocchi, to the case of singularites of infinite order. We give conditions for the possibility of this procedure in terms of local differential operators and the Gelfand-Shilov test function spaces, as well as an abstract sufficient condition. As a model case we construct a maximally positive definite state space for the Heisenberg algebra in the presence of an infinite infrared singularity. See the corresponding paper: Schmidt, Andreas U.: "Mathematical Problems of Gauge Quantum Field Theory: A Survey of the Schwinger Model" and the presentation "Infinite Infrared Regularization in Krein Spaces"
This extended write-up of a talk gives an introductory survey of mathematical problems of the quantization of gauge systems. Using the Schwinger model as an exactly tractable but nontrivial example which exhibits general features of gauge quantum field theory, I cover the following subjects: The axiomatics of quantum field theory, formulation of quantum field theory in terms of Wightman functions, reconstruction of the state space, the local formulation of gauge theories, indefiniteness of the Wightman functions in general and in the special case of the Schwinger model, the state space of the Schwinger model, special features of the model. New results are contained in the Mathematical Appendix, where I consider in an abstract setting the Pontrjagin space structure of a special class of indefinite inner product spaces - the so called quasi-positive ones. This is motivated by the indefinite inner product space structure appearing in the above context and generalizes results of Morchio and Strocchi [J. Math. Phys. 31 (1990) 1467], and Dubin and Tarski [J. Math. Phys. 7 (1966) 574]. See the corresponding paper: Schmidt, Andreas U.: "Infinite Infrared Regularization and a State Space for the Heisenberg Algebra" and the presentation "Infinite Infrared Regularization in Krein Spaces".