Interpolation of Shifted-Lacunary Polynomials

Given a “black box” function to evaluate an unknown rational polynomial f ? \mathbbQ[x]f \in {\mathbb{Q}}[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity $t \in {\mathbb{Z}}_{>0}$t \in {\mathbb{Z}}_{>0}, the shift a ? \mathbbQ\alpha \in {\mathbb{Q}}, the exponents 0 ￡ e₁ < e₂ < ? < e_t{0 \leq e_{1} < e_{2} < \cdots < e_{t}}, and the coefficients c₁, ?, c_t ? \mathbbQ \{0}c_{1}, \ldots , c_{t} \in {\mathbb{Q}} \setminus \{0\} such that

Quadratic Lower Bound for Permanent Vs. Determinant in any Characteristic

In Valiant’s theory of arithmetic complexity, the classes VP and VNP are analogs of P and NP. A fundamental problem concerning these classes is the Permanent and Determinant Problem: Given a field \mathbbF{\mathbb{F}} of characteristic ≠ 2, and an integer n, what is the minimum m such that the permanent of an n × n matrix X = (x_ij) can be expressed as a determinant of an m × m matrix, where the entries of the determinant matrix are affine linear functions of x_ij ’s, and the equality is in \mathbbF[X]{\mathbb{F}}[{\bf X}]. Mignon and Ressayre (2004) proved a quadratic lower bound m = W(n²)m = \Omega(n^{2}) for fields of characteristic 0. We extend the Mignon–Ressayre quadratic lower bound to all fields of characteristic ≠ 2.     

Tripartite entanglement sudden death in Yang-Baxter systems

In this paper, we derive unitary Yang-Baxter \breveR(q, j){\breve{R}(\theta, \varphi)} matrices from the 8×8 \mathbbM{8\times8\,\mathbb{M}} matrix and the 4 × 4 M matrix by Yang-Baxteration approach, where \mathbbM/M{\mathbb{M}/M} is the image of the braid group representation. In Yang-Baxter systems, we explore the evolution of tripartite negativity for three qubits Greenberger-Horne-Zeilinger (GHZ)-type states and W-type states and investigate the evolution of the bipartite concurrence for 2 qubits Bell-type states. We show that tripartite entanglement sudden death (ESD) and bipartite ESD all can happen in Yang-Baxter systems and find that ESD all are sensitive to the initial condition. Interestingly, we find that in the Yang-Baxter system, GHZ-type states can have bipartite entanglement and bipartite ESD, and find that in some initial conditions, W-type states have tripartite ESD while they have no bipartite Entanglement. It is worth noting that the meaningful parameter j{\varphi} has great influence on bipartite ESD for two qubits Bell-type states in the Yang-Baxter system.     

On the solution of the fuzzy Sylvester matrix equation

In this paper, we consider the fuzzy Sylvester matrix equation AX+XB=C,AX+XB=C, where A ? \mathbbR^{n ×n}A\in {\mathbb{R}}^{n \times n} and B ? \mathbbR^{m ×m}B\in {\mathbb{R}}^{m \times m} are crisp M-matrices, C is an n×mn\times m fuzzy matrix and X is unknown. We first transform this system to an (mn)×(mn)(mn)\times (mn) fuzzy system of linear equations. Then, we investigate the existence and uniqueness of a fuzzy solution to this system. We use the accelerated over-relaxation method to compute an approximate solution to this system. Some numerical experiments are given to illustrate the theoretical results.     

Characterizing sets of jobs that admit optimal greedy-like algorithms

The “Priority Algorithm” is a model of computation introduced by Borodin, Nielsen and Rackoff ((Incremental) Priority algorithms, Algorithmica 37(4):295–326, 2003) which formulates a wide class of greedy algorithms. For an arbitrary set \mathbbS\mathbb{S} of jobs, we are interested in whether or not there exists a priority algorithm that gains optimal profit on every subset of \mathbbS\mathbb{S} . In the case where the jobs are all intervals, we characterize such sets \mathbbS\mathbb{S} and give an efficient algorithm (when \mathbbS\mathbb{S} is finite) for determining this. We show that in general, however, the problem is NP-hard.     

New Results on Noncommutative and Commutative Polynomial Identity Testing

Using ideas from automata theory, we design the first polynomial deterministic identity testing algorithm for the sparse noncommutative polynomial identity testing problem. Given a noncommuting black-box polynomial f ? \mathbb F{x₁,?,x_n}f \in {\mathbb F}\{x_{1},\ldots,x_n\} of degree d with at most t monomials, where the variables x_i are noncommuting, we give a deterministic polynomial identity test that checks if C o 0C \equiv 0 and runs in time polynomial in d, n, |C|, and t. Our algorithm evaluates the black-box polynomial for x_i assigned to matrices over \mathbbF{\mathbb{F}} and, in fact, reconstructs the entire polynomial f in time polynomial in n, d and t.     

Caching Is Hard—Even in the Fault Model

We prove strong \mathbb NP{\mathbb {NP}}-completeness for the four variants of caching with multi-size pages. These four variants are obtained by choosing either the fault cost or the bit cost model, and by combining it with either a forced or an optional caching policy. This resolves two questions in the area of paging and caching that were open since the 1990s.     

Entanglement and Berry phase in a 9 × 9 Yang–Baxter system

A M-matrix which satisfies the Hecke algebraic relations is presented. Via the Yang–Baxterization approach, we obtain a unitary solution \breveR(q,j₁,j₂){\breve{R}(\theta,\varphi_{1},\varphi_{2})} of Yang–Baxter equation. It is shown that any pure two-qutrit entangled states can be generated via the universal \breveR{\breve{R}}-matrix assisted by local unitary transformations. A Hamiltonian is constructed from the \breveR{\breve{R}}-matrix, and Berry phase of the Yang–Baxter system is investigated. Specifically, for j₁ = j₂{\varphi_{1}\,{=}\,\varphi_{2}}, the Hamiltonian can be represented based on three sets of SU(2) operators, and three oscillator Hamiltonians can be obtained. Under this framework, the Berry phase can be interpreted.     

VPSPACE and a Transfer Theorem over the Reals

We introduce a new class VPSPACE of families of polynomials. Roughly speaking, a family of polynomials is in VPSPACE if its coefficients can be computed in polynomial space. Our main theorem is that if (uniform, constant-free) VPSPACE families can be evaluated efficiently then the class \sf PAR_{\mathbb R}\sf {PAR}_{\mathbb {R}} of decision problems that can be solved in parallel polynomial time over the real numbers collapses to \sfP_{\mathbb R}\sf{P}_{\mathbb {R}}. As a result, one must first be able to show that there are VPSPACE families which are hard to evaluate in order to separate \sfP_{\mathbb R}\sf{P}_{\mathbb {R}} from \sfNP_{\mathbb R}\sf{NP}_{\mathbb {R}}, or even from \sfPAR_{\mathbb R}\sf{PAR}_{\mathbb {R}}.     

Automata theory based on lattice-ordered semirings

In this paper, definitions of K{\mathcal{K}} automata, K{\mathcal{K}} regular languages, K{\mathcal{K}} regular expressions and K{\mathcal{K}} regular grammars based on lattice-ordered semirings are given. It is shown that K{\mathcal{K}}NFA is equivalent to K{\mathcal{K}}DFA under some finite condition, the Pump Lemma holds if K{\mathcal{K}} is finite, and Ke{{\mathcal{K}}}\epsilonNFA is equivalent to K{\mathcal{K}}NFA. Further, it is verified that the concatenation of K{\mathcal{K}} regular languages remains a K{\mathcal{K}} regular language. Similar to classical cases and automata theory based on lattice-ordered monoids, it is also found that K{\mathcal{K}}NFA, K{\mathcal{K}} regular expressions and K{\mathcal{K}} regular grammars are equivalent to each other when K{\mathcal{K}} is a complete lattice.     

Mutually unbiased special entangled bases with Schmidt number 2 in $${\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}$$

A way of constructing special entangled basis with fixed Schmidt number 2 (SEB2) in \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}(k\in z^+,3\not \mid k)\) is proposed, and the conditions mutually unbiased SEB2s (MUSEB2s) satisfy are discussed. In addition, a very easy way of constructing MUSEB2s in \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}(k=2^l)\) is presented. We first establish the concrete construction of SEB2 and MUSEB2s in \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{4}\) and \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{8}\), respectively, and then generalize them into \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}(k\in z^+,3\not \mid k)\) and display the condition that MUSEB2s satisfy; we also give general form of two MUSEB2s as examples in \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}(k=2^l)\).     

Lower Bounds for Agnostic Learning via Approximate Rank

We prove that the concept class of disjunctions cannot be pointwise approximated by linear combinations of any small set of arbitrary real-valued functions. That is, suppose that there exist functions f₁, ?, f_r\phi_{1}, \ldots , \phi_{r} : {− 1, 1}ⁿ → \mathbbR{\mathbb{R}} with the property that every disjunction f on n variables has $\|f - \sum\nolimits_{i=1}^{r} \alpha_{i}\phi _{i}\|_{\infty}\leq 1/3$\|f - \sum\nolimits_{i=1}^{r} \alpha_{i}\phi _{i}\|_{\infty}\leq 1/3 for some reals a₁, ?, a_r\alpha_{1}, \ldots , \alpha_{r}. We prove that then $r \geq exp \{\Omega(\sqrt{n})\}$r \geq exp \{\Omega(\sqrt{n})\}, which is tight. We prove an incomparable lower bound for the concept class of decision lists. For the concept class of majority functions, we obtain a lower bound of W(2ⁿ/n)\Omega(2^{n}/n) , which almost meets the trivial upper bound of 2ⁿ for any concept class. These lower bounds substantially strengthen and generalize the polynomial approximation lower bounds of Paturi (1992) and show that the regression-based agnostic learning algorithm of Kalai et al. (2005) is optimal.     

A Representation of the Interval Hull of a Tolerance Polyhedron Describing Inclusions of Function Values and Slopes

Given a nonempty set of functions

On the efficiency of quantum algorithms for Hamiltonian simulation

We study algorithms simulating a system evolving with Hamiltonian H = ?_j=1^m H_j{H = \sum_{j=1}^m H_j} , where each of the H _j , j = 1, . . . ,m, can be simulated efficiently. We are interested in the cost for approximating e^-iHt, t ? \mathbbR{e^{-iHt}, t \in \mathbb{R}} , with error e{\varepsilon} . We consider algorithms based on high order splitting formulas that play an important role in quantum Hamiltonian simulation. These formulas approximate e ^−iHt by a product of exponentials involving the H _j , j = 1, . . . ,m. We obtain an upper bound for the number of required exponentials. Moreover, we derive the order of the optimal splitting method that minimizes our upper bound. We show significant speedups relative to previously known results.     

A Dynamic Logic of Agency II: Deterministic $${\mathcal{DLA}}$$ , Coalition Logic,and Game Theory

We continue the work initiated in Herzig and Lorini (J Logic Lang Inform, in press) whose aim is to provide a minimalistic logical framework combining the expressiveness of dynamic logic in which actions are first-class citizens in the object language, with the expressiveness of logics of agency such as STIT and logics of group capabilities such as CL and ATL. We present a logic called DDLA{\mathcal{DDLA}} (Deterministic Dynamic logic of Agency) which supports reasoning about actions and joint actions of agents and coalitions, and agentive and coalitional capabilities. In DDLA{\mathcal{DDLA}} it is supposed that, once all agents have selected a joint action, the effect of this joint action is deterministic. In order to assess DDLA{\mathcal{DDLA}} we prove that it embeds Coalition Logic. We then extend DDLA{\mathcal{DDLA}} with modal operators for agents’ preferences, and show that the resulting logic is sufficiently expressive to capture the game-theoretic concepts of best response and Nash equilibrium.     

Spigot Algorithm and Reliable Computation of Natural Logarithm

The spigot approach used in the previous paper (Reliable Computing 7 (3) (2001), pp. 247–273) for root computation is now applied to natural logarithms. The logarithm ln Q with Q , Q > 1 is decomposed into a sum of two addends k ₁× ln Q ₁+k ₂× ln Q ₂ with k ₁, k ₂ , then each of them is computed by the spigot algorithm and summation is carried out using integer arithmetic. The whole procedure is not literally a spigot algorithm, but advantages are the same: only integer arithmetic is needed whereas arbitrary accuracy is achieved and absolute reliability is guaranteed. The concrete procedure based on the decomposition with p, q ( – {0}), p < q is simple and ready for implementation. In addition to the mentioned paper, means for determining an upper bound for the biggest integer occurring in the process of spigot computing are now provided, which is essential for the reliability of machine computation.     

The local distinguishability of any three generalized Bell states

We study the problem of distinguishing maximally entangled quantum states by using local operations and classical communication (LOCC). A question of fundamental interest is whether any three maximally entangled states in \({\mathbb {C}}^d\otimes {\mathbb {C}}^d (d\ge 4)\) are distinguishable by LOCC. In this paper, we restrict ourselves to consider the generalized Bell states. And we prove that any three generalized Bell states in \({\mathbb {C}}^d\otimes {\mathbb {C}}^d (d\ge 4)\) are locally distinguishable.     

Efficient temporal counting with bounded error

This paper studies aggregate search in transaction time databases. Specifically, each object in such a database can be modeled as a horizontal segment, whose y-projection is its search key, and its x-projection represents the period when the key was valid in history. Given a query timestamp q _t and a key range , a count-query retrieves the number of objects that are alive at q _t , and their keys fall in . We provide a method that accurately answers such queries, with error less than , where N _alive(q _t ) is the number of objects alive at time q _t , and ɛ is any constant in (0, 1]. Denoting the disk page size as B, and n =  N / B, our technique requires O(n) space, processes any query in O(log _B n) time, and supports each update in O(log _B n) amortized I/Os. As demonstrated by extensive experiments, the proposed solutions guarantee query results with extremely high precision (median relative error below 5%), while consuming only a fraction of the space occupied by the existing approaches that promise precise results.     

The Gaussian Surface Area and Noise Sensitivity of Degree-d Polynomial Threshold Functions

We prove asymptotically optimal bounds on the Gaussian noise sensitivity and Gaussian surface area of degree-d polynomial threshold functions. In particular, we show that for f a degree-d polynomial threshold function that the Gaussian noise sensitivity of f with parameter e{\epsilon} is at most \fracdarcsin(?{2e-e²})p{\frac{d\arcsin\left(\sqrt{2\epsilon-\epsilon^2}\right)}{\pi}} . This bound translates into an optimal bound on the Gaussian surface area of such functions, namely that the Gaussian surface area is at most \fracd?{2p}{\frac{d}{\sqrt{2\pi}}} . Finally, we note that the later result implies bounds on the runtime of agnostic learning algorithms for polynomial threshold functions.