Varieties Characterized by their Endomorphisms

We show that two varieties X and Y with isomorphic endomorphism semigroups are isomorphic up to field automorphism if one of them is affine and contains a copy of the affine line. A holomorphic version of this result is due to the first author.Comment: 8 page

Recommender Systems with Characterized Social Regularization

Social recommendation, which utilizes social relations to enhance recommender systems, has been gaining increasing attention recently with the rapid development of online social network. Existing social recommendation methods are based on the fact that users preference or decision is influenced by their social friends' behaviors. However, they assume that the influences of social relation are always the same, which violates the fact that users are likely to share preference on diverse products with different friends. In this paper, we present a novel CSR (short for Characterized Social Regularization) model by designing a universal regularization term for modeling variable social influence. Our proposed model can be applied to both explicit and implicit iteration. Extensive experiments on a real-world dataset demonstrate that CSR significantly outperforms state-of-the-art social recommendation methods.Comment: to appear in CIKM 201

FPT is Characterized by Useful Obstruction Sets

Many graph problems were first shown to be fixed-parameter tractable using the results of Robertson and Seymour on graph minors. We show that the combination of finite, computable, obstruction sets and efficient order tests is not just one way of obtaining strongly uniform FPT algorithms, but that all of FPT may be captured in this way. Our new characterization of FPT has a strong connection to the theory of kernelization, as we prove that problems with polynomial kernels can be characterized by obstruction sets whose elements have polynomial size. Consequently we investigate the interplay between the sizes of problem kernels and the sizes of the elements of such obstruction sets, obtaining several examples of how results in one area yield new insights in the other. We show how exponential-size minor-minimal obstructions for pathwidth k form the crucial ingredient in a novel OR-cross-composition for k-Pathwidth, complementing the trivial AND-composition that is known for this problem. In the other direction, we show that OR-cross-compositions into a parameterized problem can be used to rule out the existence of efficiently generated quasi-orders on its instances that characterize the NO-instances by polynomial-size obstructions.Comment: Extended abstract with appendix, as accepted to WG 201

On TT-characterized subgroups of compact Abelian groups

We say that a subgroup $H$ of an infinite compact Abelian group $X$ is {\it $T$-characterized} if there is a $T$-sequence $\mathbf{u} =\{u_n \}$ in the dual group of $X$ such that $H=\{x\in X: \; (u_n, x)\to 1 \}$. We show that a closed subgroup $H$ of $X$ is $T$-characterized if and only if $H$ is a $G_\delta$-subgroup of $X$ and the annihilator of $H$ admits a Hausdorff minimally almost periodic group topology. All closed subgroups of an infinite compact Abelian group $X$ are $T$-characterized if and only if $X$ is metrizable and connected. We prove that every compact Abelian group $X$ of infinite exponent has a $T$-characterized subgroup which is not an $F_{\sigma}$-subgroup of $X$ that gives a negative answer to Problem 3.3 in [10]

Quasirandom permutations are characterized by 4-point densities

For permutations π and τ of lengths |π|≤|τ| , let t(π,τ) be the probability that the restriction of τ to a random |π| -point set is (order) isomorphic to π . We show that every sequence {τj} of permutations such that |τj|→∞ and t(π,τj)→1/4! for every 4-point permutation π is quasirandom (that is, t(π,τj)→1/|π|! for every π ). This answers a question posed by Graham

Quantum State Transfer Characterized by Mode Entanglement

We study the quantum state transfer (QST) of a class of tight-bonding Bloch electron systems with mirror symmetry by considering the mode entanglement. Some rigorous results are obtained to reveal the intrinsic relationship between the fidelity of QST and the mirror mode concurrence (MMC), which is defined to measure the mode entanglement with a certain spatial symmetry and is just the overlap of a proper wave function with its mirror image. A complementarity is discovered as the maximum fidelity is accompanied by a minimum of MMC. And at the instant, which is just half of the characteristic time required to accomplish a perfect QST, the MMC can reach its maximum value one. A large class of perfect QST models with a certain spectrum structure are discovered to support our analytical results.Comment: 6 pages, 3 figures. to appear in PR

Every locally characterized affine-invariant property is testable

Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-invariant property having local characterizations are testable. In fact, we show a proximity-oblivious test for any such property P, meaning that there is a test that, given an input function f, makes a constant number of queries to f, always accepts if f satisfies P, and rejects with positive probability if the distance between f and P is nonzero. More generally, we show that any affine-invariant property that is closed under taking restrictions to subspaces and has bounded complexity is testable. We also prove that any property that can be described as the property of decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-d polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results depend on a new Gowers inverse theorem by Tao and Ziegler for low characteristic fields that decomposes any polynomial with large Gowers norm into a function of low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties