Solution Regularity of k-partite Linear Systems -- Variant of Rado's Theorem

Ramsey theory is a modern and arresting field of mathematics with a century of history. The basic paradigm of Ramsey Theory is that any sufficiently large regular structure contains some highly ordered substructure. One number theoretic Ramseyan result is the Rado's Theorem. In this research, I extended the Rado's Theorem to a more general model. Given $k \ge 2$, there are some necessary and sufficient conditions such that for a system of linear equations to be semi-regular on $\mathbb{Z}$. In this article, I will show the proof to this extended theorem with a combination of conventional strategies (applied to prove Rado's Theorem) and new perspectives (in linear algebra and number theory). Some polynomials methods are applied here to make use of the existence of infinitely many desired prime numbers.Comment: 18 pages (including table of contents and references

Numerical Framework for Semi-Device-Independent Quantum Random Number Generators

Quantum random number generator (QRNG) is one of the most widely applied branches in quantum cryptography. Among all QRNG schemes, semi-device-independent (semi-DI) QRNG is quite promising, achieving high randomness generation rate with few assumptions on the devices. For the central task of a QRNG study -- security analysis, numerical approaches become popular for its generality to various semi-DI QRNG schemes. Here we formulate a numerical framework for the finite-size security of general semi-DI QRNGs, which gives a secure lower bound of the finite-size randomness generation rate against general attacks. We consider a simple example of an optical semi-DI QRNG as an application of our framework.Comment: 8 pages, 4 figure

Source-independent quantum random number generation

Quantum random number generators can provide genuine randomness by appealing to the fundamental principles of quantum mechanics. In general, a physical generator contains two parts---a randomness source and its readout. The source is essential to the quality of the resulting random numbers; hence, it needs to be carefully calibrated and modeled to achieve information-theoretical provable randomness. However, in practice, the source is a complicated physical system, such as a light source or an atomic ensemble, and any deviations in the real-life implementation from the theoretical model may affect the randomness of the output. To close this gap, we propose a source-independent scheme for quantum random number generation in which output randomness can be certified, even when the source is uncharacterized and untrusted. In our randomness analysis, we make no assumptions about the dimension of the source. For instance, multiphoton emissions are allowed in optical implementations. Our analysis takes into account the finite-key effect with the composable security definition. In the limit of large data size, the length of the input random seed is exponentially small compared to that of the output random bit. In addition, by modifying a quantum key distribution system, we experimentally demonstrate our scheme and achieve a randomness generation rate of over $5\times 10^3$ bit/s.Comment: 11 pages, 7 figure

Unified framework for quantumness -- coherence, discord, and entanglement

From an operational perspective, quantumness characterizes the exotic behavior in a physical process which cannot be explained with Newtonian physics. There are several widely used measures of quantumness, including coherence, discord, and entanglement, each proven to be essential resources in particular situations. There exists evidence of fundamental connections amongst the three measures. However, those quantumnesses are still regarded differently and such connections are yet to be elucidated. Here, we introduce a general framework of defining a unified quantumness with an operational motivation founded on the capability of interferometry. The quantumness appears differently as coherence, discord, and entanglement in different scenarios with local measurement, weak reference frame free measurement, and strong reference frame free measurement, respectively. Our results also elaborate how these three measures are related and how they can be transformed from each other. This framework can be further extended to other scenarios and serves as a universal quantumness measure.Comment: 9 pages, 4 figure

Quantifying the dissipation enhancement of cellular flows

We study the dissipation enhancement by cellular flows. Previous work by Iyer, Xu, and Zlato\v{s} produces a family of cellular flows that can enhance dissipation by an arbitrarily large amount. We improve this result by providing quantitative bounds on the dissipation enhancement in terms of the flow amplitude, cell size and diffusivity. Explicitly we show that the mixing time is bounded by the exit time from one cell when the flow amplitude is large enough, and by the reciprocal of the effective diffusivity when the flow amplitude is small. This agrees with the optimal heuristics. We also prove a general result relating the dissipation time of incompressible flows to the mixing time. The main idea behind the proof is to study the dynamics probabilistically and construct a successful coupling.Comment: 21 pages, 2 figure

Web-based toolkits for topology prediction of transmembrane helical proteins, fold recognition, structure and binding scoring, folding-kinetics analysis and comparative analysis of domain combinations

We have developed the following web servers for protein structural modeling and analysis at http:// theory.med.buffalo.edu: THUMBUP, UMDHMMTMHP and TUPS, predictors of trans-membrane helical protein topology based on a mean-burial-propensity scale of amino acid residues (THUMBUP), hidden Markov model (UMDHMMTMHP) and their combinations (TUPS); SPARKS 2.0 and SP3, two profile– profile alignment methods, that match input query sequence(s) to structural templates by integrating sequence profile with knowledge-based structural score (SPARKS 2.0) and structure-derived profile (SP3); DFIRE, a knowledge-based potential for scoring free energy of monomers (DMONOMER), loop conformations (DLOOP), mutant stability (DMUTANT) and binding affinity of protein–protein/ peptide/DNA complexes (DCOMPLEX & DDNA); TCD, a program for protein-folding rate and transition-state analysis of small globular proteins; and DOGMA, a web-server that allows comparative analysis of domain combinations between plant and other 55 organisms. These servers provide tools for prediction and/or analysis of proteins on the secondary structure, tertiary structure and interaction levels, respectively

Web-based toolkits for topology prediction of transmembrane helical proteins, fold recognition, structure and binding scoring, folding-kinetics analysis and comparative analysis of domain combinations

We have developed the following web servers for protein structural modeling and analysis at : THUMBUP, UMDHMM(TMHP) and TUPS, predictors of transmembrane helical protein topology based on a mean-burial-propensity scale of amino acid residues (THUMBUP), hidden Markov model (UMDHMM(TMHP)) and their combinations (TUPS); SPARKS 2.0 and SP(3), two profile–profile alignment methods, that match input query sequence(s) to structural templates by integrating sequence profile with knowledge-based structural score (SPARKS 2.0) and structure-derived profile (SP(3)); DFIRE, a knowledge-based potential for scoring free energy of monomers (DMONOMER), loop conformations (DLOOP), mutant stability (DMUTANT) and binding affinity of protein–protein/peptide/DNA complexes (DCOMPLEX & DDNA); TCD, a program for protein-folding rate and transition-state analysis of small globular proteins; and DOGMA, a web-server that allows comparative analysis of domain combinations between plant and other 55 organisms. These servers provide tools for prediction and/or analysis of proteins on the secondary structure, tertiary structure and interaction levels, respectively